Real time feedback-based optimization of distributed energy resources

ABSTRACT

An example device includes a processor configured to receive a plurality of voltage values representing respective voltage magnitudes at voltage nodes in a first portion of a power system and determine, for each voltage node, a respective value of first and second voltage-constraint coefficients. The processor is also configured to receive a power value corresponding to a connection point of the first portion of the power system with a second portion of the power system and determine for the connection point, a respective value of first and second power-constraint coefficients. The processor is also configured to cause at least one energy resource connected to the first portion of the power system to modify an output power of the at least one energy resource based on the value of the first and second voltage-constraint coefficients for each voltage node and the value of the first and second power-constraint coefficients.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.15/814,532, filed Nov. 16, 2017, which claims the benefit of U.S.Provisional Application No. 62/422,853, titled “DISTRIBUTIONINFRASTRUCTURE OPTIMIZATION AND CONTROL” and filed Nov. 16, 2016, andU.S. Provisional Application No. 62/567,628, titled “DISTRIBUTIONINFRASTRUCTURE OPTIMIZATION AND CONTROL” and filed Oct. 3, 2017. Theentire content of each listed application is incorporated herein byreference.

CONTRACTUAL ORIGIN

The United States Government has rights in this invention under ContractNo. DE-AC36-08GO28308 between the United States Department of Energy andAlliance for Sustainable Energy, LLC, the Manager and Operator of theNational Renewable Energy Laboratory.

BACKGROUND

As renewable energy becomes more important in today's society, powergrids may have to manage increasingly distributed energy resources. Evenmodest housing may have photovoltaic (PV) systems and/or wind turbinesinstalled to reduce dependence on the grid, and to offset energy costs.As prevalence of these distributed energy resources increases, gridmanagers, such as those who manage power distribution networks, will befaced with new challenges in preventing network overload, ensuringsatisfaction of engineering limits, and managing power generated bydistributed sources.

SUMMARY

The present disclosure provides systems, devices, and methods for powersystems such as power distribution grids to manage distributed energyresources (DERs). The techniques described herein may allow adistribution feeder or other power system to emulate a “virtual powerplant,” effectively providing automatic generation control (AGC) and/orregulation services with respect to particular points in the powersystem, such as at an interface with a power transmission grid, whileaddressing DER owners' economic objectives and ensuring satisfaction ofengineering limits.

In one example, a device includes at least one processor configured toreceive a plurality of voltage values, wherein voltage values in theplurality of voltage values correspond to respective voltage nodes in aplurality of voltage nodes in a first portion of a power system. The atleast one processor is also configured to determine, for each respectivevoltage node, a respective value of a first voltage-constraintcoefficient, based on a respective previous value of the firstvoltage-constraint coefficient, a minimum voltage value, and arespective voltage value in the plurality of voltage values thatcorresponds to the respective voltage node, and a respective value of asecond voltage-constraint coefficient based on a respective previousvalue of the second voltage-constraint coefficient, a maximum voltagevalue, and the respective voltage value. The at least one processor isalso configured to receive a power value corresponding to a connectionpoint of the first portion of the power system with a second portion ofthe power system and determine for the connection point, a value of afirst power-constraint coefficient, based on a previous value of thefirst power-constraint coefficient, a power setpoint for the connectionpoint, and the power value, and a value of a second power-constraintcoefficient based on a previous value of the second voltage-constraintcoefficient, the power setpoint for the connection point, and the powervalue. The at least one processor is also configured to cause at leastone energy resource in a plurality of energy resources that areconnected to the first portion of the power system to modify an outputpower of the at least one energy resource based on the respective valueof the first voltage-constraint coefficient for each respective voltagenode, the respective value of the second voltage-constraint coefficientfor each respective voltage node, the value of the firstpower-constraint coefficient for the connection point, and the value ofthe second power-constraint coefficient for the connection point.

In another example, a system includes a power management systemconfigured to receive, from each of a plurality of voltage measurementdevices, a respective voltage value that corresponds to a respectivevoltage node in a plurality of voltage nodes in a first portion of apower system and receive a power value that corresponds to a connectionpoint at which the first portion of the power system connects to asecond portion of the power system. The power management system isfurther configured to determine, for each respective voltage node in theplurality of voltage nodes, a respective value of a firstvoltage-constraint coefficient, based on a respective previous value ofthe first voltage-constraint coefficient, a minimum voltage value, andthe respective voltage value, and a respective value of a secondvoltage-constraint coefficient based on a respective previous value ofthe second voltage-constraint coefficient, a maximum voltage value, andthe respective voltage value. The power management system is furtherconfigured to determine, for the connection point, a value of a firstpower-constraint coefficient, based on a previous value of the firstpower-constraint coefficient, a power setpoint for the connection point,and the power value, and a value of a second power-constraintcoefficient, based on a previous value of the second power-constraintcoefficient, the power setpoint for the connection point, and the powervalue. The power management system is further configured to output therespective value of the first voltage-constraint coefficient for eachrespective voltage node, the respective value of the secondvoltage-constraint coefficient for each respective voltage node, thevalue of the first power-constraint coefficient, and the value of thesecond power-constraint coefficient. The system further includes aplurality of energy resource management devices, each corresponding to arespective at least one energy resource connected to the power system,each energy resource management device being configured to receive therespective value of the first voltage-constraint coefficient for eachrespective voltage node, the respective value of the secondvoltage-constraint coefficient for each respective voltage node, thevalue of the first power-constraint coefficient, and the value of thesecond power-constraint coefficient. Each energy resource managementdevice is further configured to determine, based on the respective valueof the first voltage-constraint coefficient for each respective voltagenode, the respective value of the second voltage-constraint coefficientfor each respective voltage node, the value of the firstpower-constraint coefficient, and the value of the secondpower-constraint coefficient, a respective power setpoint value, andmodify a respective output power of the respective at least one energyresource, based on the respective power setpoint value.

In another example, a method includes receiving, by a power managementsystem comprising at least one processor, a plurality of voltage values,wherein voltage values in the plurality of voltage values correspond torespective voltage nodes in a plurality of voltage nodes in a firstportion of a power system. The method also includes determining, by thepower management system and for each respective voltage node, arespective value of a first voltage-constraint coefficient, based on arespective previous value of the first voltage-constraint coefficient, aminimum voltage value, and a respective voltage value in the pluralityof voltage values that corresponds to the respective voltage node, and arespective value of a second voltage-constraint coefficient based on arespective previous value of the second voltage-constraint coefficient,a maximum voltage value, and the respective voltage value. The methodalso includes receiving, by the power management system, a power valuecorresponding to a connection point of the first portion of the powersystem with a second portion of the power system and determining, by thepower management system and for the connection point, a value of a firstpower-constraint coefficient, based on a previous value of the firstpower-constraint coefficient, a power setpoint for the connection point,and the power value, and a value of a second power-constraintcoefficient based on a previous value of the second voltage-constraintcoefficient, the power setpoint for the connection point, and the powervalue. The method also includes causing, by the power management system,at least one energy resource in a plurality of energy resources that areconnected to the first portion of the power system to modify an outputpower of the at least one energy resource based on the respective valueof the first voltage-constraint coefficient for each respective voltagenode, the respective value of the second voltage-constraint coefficientfor each respective voltage node, the value of the firstpower-constraint coefficient for the connection point, and the value ofthe second power-constraint coefficient for the connection point.

The details of one or more examples are set forth in the accompanyingdrawings and the description below. Other features, objects, andadvantages will be apparent from the description and drawings, and fromthe claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a conceptual diagram illustrating an example power managementsystem configured to manage distributed energy resources in a powersystem, in accordance with one or more aspects of the presentdisclosure.

FIG. 2 is a conceptual diagram illustrating another example powermanagement system configured to manage distributed energy resources in apower system, in accordance with one or more aspects of the presentdisclosure.

FIG. 3 is a flow diagram illustrating example operations for performingreal time feedback-based optimization of distributed energy resources,in accordance with one or more aspects of the present disclosure.

FIG. 4 is a flow diagram illustrating example operations for performingreal time feedback-based optimization of distributed energy resources,in accordance with one or more aspects of the present disclosure.

FIG. 5 is a conceptual diagram illustrating another example powermanagement system configured to manage distributed energy resources in apower system, in accordance with one or more aspects of the presentdisclosure.

DETAILED DESCRIPTION

The present disclosure may provide systems, devices, and methods forreal-time (or near-real-time) regulation of energy resources in a powerdistribution grid or other power system to maximize operationalobjectives. The techniques described herein may, for example, beemployed in operation and control of power systems having highintegration of distributed energy resources (DERs). Voltage, power, andcurrent measurements from locations in the power system may be collectedand used to update a set of coefficients. Those coefficients may then beused to update power setpoints for DERs in the power system.

Related art approaches for regulating frequency and maintaining areliable operation of transmission systems may leverage primaryfrequency response, AGC, and/or regulation services provided bylarge-scale synchronous generators. In the future, however, DERs at bothutility and residential/commercial levels will likely need to supplementgeneration-side capabilities, by providing additional flexibility inregulating frequency and maintaining reliable system operation.

The techniques of the present disclosure may be implemented in variousdevices to achieve such flexibility. By utilizing multiple measurementtypes from throughout the power system, the techniques described hereinmay provide improved frequency regulation, more reliable systemoperation, and/or improved power generation and/or consumption.Furthermore, the techniques described herein address both wye- anddelta-connections, allowing for implementation in power systems havingmulti-phase devices as well as power systems having single phasedevices. In addition, the techniques of the present disclosure mayprovide a uniform approach that addresses collections of DERs, which maybe controlled as a group, as well as single, individually-controlledDERs.

FIG. 1 is a conceptual diagram illustrating an example power managementsystem (e.g., system 2) configured to manage energy resources in a powersystem, in accordance with one or more aspects of the presentdisclosure. In the example of FIG. 1, system 2 includes power managementunit 4 and control devices 10A and 10B (collectively “control devices10”). System 2 also includes nodes 6A-6C (collectively “nodes 6”),connection point 7, and energy resources 8A-8C (collectively “energyresources 8”). As shown in the example of FIG. 1, nodes 6, connectionpoint 7, and control devices 10 are all connected via a network of powerlines and, with those power lines, may represent a “power system”.

System 2, as shown in the example of FIG. 1, manages a simplified powersystem. In other examples, the power system may include any number ofadditional ones of nodes 6, energy resources 8, and/or control devices10. Thus, while shown in FIG. 1 as having three nodes and four energyresources, the power system may, in other examples, include more orfewer nodes, energy resources, and/or control devices in other examples.For instance, the techniques of the present disclosure may be used witha micro-grid, a subset of a power distribution network, an entire powerdistribution network, a community power grid (e.g., in which acollection of residents implement a local power network), a campus powergrid (e.g., in which a company or educational institution implements itsown power network), or any other collection of connected powergeneration and consumption devices. Additionally, System 2 of FIG. 1represents only one example of a system configured to perform thetechniques described herein, and various other systems, havingadditional components, fewer components, and/or other components, may beused in accordance with the present disclosure.

In the example of FIG. 1, nodes 6 are devices configured to measureelectrical quantities at a location of the power system and output themeasurement. For example, nodes 6 may be configured to measure voltagevalues and/or current values at their respective locations. Nodes 6 maybe located at any point in the power system. In some examples, one ormore of nodes 6 may be in the middle of a power line. In some examples,one or more of nodes 6 may be at a junction of two or more power lines.Examples of nodes 6 include phasor measurement units, current meters,inverters, power substations, and other systems or devices capable ofdetermining the relevant electrical quantity at a location in the powersystem.

Nodes 6A and 6B are configured to determine respective voltagemeasurements 12A and 12B on an iterative basis. Each voltage measurementmay represent the present voltage at a particular location in the powersystem. Node 6C is configured to determine a current measurement 12C onan iterative basis. Each current measurement may represent the presentcurrent flowing through a particular power line in the power system.Voltage measurements 12A and 12B and current measurement 12C may becollectively referred to herein as “measurements 12”.

In the example of FIG. 1, connection point 7 represents a point at whichthe power system is connected to a larger system. For example, the powersystem shown in FIG. 1 may represent a power distribution network andconnection point 7 may represent its connection to a power transmissionnetwork. As another example, the power system in FIG. 1 may represent asubset of a power distribution network and connection point 7 mayconnect the power system to the broader power distribution network. Inother words, connection point 7 is the point at which the smaller powersystem shown in FIG. 1 connects to the rest of the power system.

Connection point 7 is configured to measure a local power measurementand output that measurement. In the example of FIG. 1, for instanceconnection point 7 is configured to determine power measurement 13 on aniterative basis. Power measurement 13 may represent an amount of realpower being delivered to the power network shown in FIG. 1 from the restof the power network. Examples of connection point 7 include a powersubstation, a point of connection of a microgrid to the rest of thegrid, a single metering point for a community-level aggregation, orother suitable facility and/or device.

Components of system 2 (e.g., nodes 6, connection point 7, powermanagement unit 4, and/or control devices 10) may be configured toperform the techniques described herein in an iterative fashion thatallows system 2 to maximize operational objectives while coping with thevariability of ambient conditions and non-controllable assets within thepower system. That is, the techniques described herein may be performedon a relatively fast time scale, thereby allowing more efficientoperation of the power system while ensuring that physical constraints(e.g., line maximums, device safety standards, etc.) are maintained. Forinstance, the components of system 2 may perform operations everysecond, every millisecond, or at some other interval. In some examples,different components may perform operations at different intervals whilein other examples, all components of system 2 may generally perform theoperations described herein with the same frequency.

In accordance with the techniques described herein, nodes 6 may outputmeasurements 12 and connection point 7 may output power measurement 13.In the example of FIG. 1, for instance, nodes 6 and connection point 7may transmit measurements 12 and power measurement 13 to powermanagement unit 4 using wireless and/or wired communication. In otherexamples, nodes 6 and/or connection point 7 may additionally oralternatively transmit measurements 12 and/or power measurement 13 toone or more other components of system 2, such as one or more of controldevices 10.

In the example of FIG. 1, power management unit 4 is configured tomanage the power system shown in FIG. 1 to provide power to consumers.In such example, power management unit 4 may manage the distribution ofpower from DERs within the power system shown in FIG. 1, as well as thereceipt and distribution of power from the larger power system (e.g.,via connection point 7), while avoiding overloading and ensuring thatconsumers' power needs are met. In some examples, power management unit4 may represent a system owned and operated by a utility company. Inother examples, power management unit 4 may be owned and/or operated byanother entity. For instance, power management unit 4 may represent anaccess point of a power network of a business park or corporate campus.As another example, power management unit 4 may manage a micro-grid,such as may be employed on a military base, mobile hospital, or othersmall area in which electrical power may be desirable. In other words,power management unit 4 may represent any system configured to managepower distribution via a power network.

Power management unit 4 may be a computing device, such as a servercomputer, a desktop computer, or any other device capable ofimplementing some or all of the techniques described herein. In someexamples, power management unit 4 may represent a cloud computingenvironment. That is, while shown as a single box in the example of FIG.1, power management unit 4 may, in some examples, be a group ofdistributed computing resources that communicate with one another toperform at least some of the techniques described herein. In someexamples, power management unit 4 may be the same as or be physicallycollocated with connection point 7. For instance, connection point 7 mayrepresent the connection between the power system shown in FIG. 1 and apower transmission network and may be a power substation that isconfigured to perform the operations of power management unit 4 asdescribed herein. In some examples, such as the example shown in FIG. 1,connection point 7 and power management unit 4 may be physicallyseparated.

In the example of FIG. 1, power management unit 4 may receivemeasurements 12 and power measurement 13. Based on measurements 12 andpower measurement 13, power management unit 4 may iteratively determinea set of coefficient values (“coefficient values 14”). Thesecoefficients may be related to defined voltage, current, and powerconstraints in the power system, and may be determined as furtherdescribed herein. In the simplified example of FIG. 1, for instance,power management unit 4 may determine two voltage-constraintcoefficients for each node from which a voltage measurement is received,a current-constraint coefficient for each node from which a currentmeasurement is received, and two power-constraint coefficients. Thus, inthe simplified example of FIG. 1, coefficient values 14 may includeseven values.

For each of nodes 6A and 6B, power management unit 4 may determine afirst voltage-constraint coefficient value based on a previous value ofthe first voltage-constraint coefficient for the node, a minimum voltagevalue for the node, and the voltage measurement for the node. Thus, fornode 6A, power management unit 4 may determine a firstvoltage-constraint coefficient value based on the previous firstvoltage-constraint coefficient value for node 6A, a minimum voltagevalue, and voltage measurement 12A. Similarly, for each of nodes 6A and6B, power management unit 4 may determine a second voltage-constraintcoefficient value based on a previous value of the firstvoltage-constraint coefficient, a maximum voltage value, and the voltagemeasurement for the node. In some examples, the first and secondvoltage-constraint coefficient values for each node may be determinedadditionally or alternatively based on other criteria. Determination ofvalues for the first and second voltage-constraint coefficients isfurther described with respect to FIGS. 2-4 below. Power management unit4, in the example of FIG. 1, may output coefficient values 14 to each ofcontrol devices 10.

Control devices 10, in the example of FIG. 1, are configured to managethe power output of one or more respective energy resources. Forinstance, inverter 10A may represent an inverter configured to receivepower from energy resource 8A (e.g., a PV panel) and transform the powerinto a form that can be transmitted via the connected power system.Power inverters, in general, may perform various operations to make thepower output of energy resources more stable and/or more compatible withpower systems. Control device 10B may represent a home energy managementdevice configured to receive power from energy resources 8B, 8C, and 8D(e.g., a PV panel, an energy storage device, and an electric vehicle(EV), respectively), manage the distribution of that power among thehome, and manage power received or transmitted via the connected powersystem. That is, control device 10B may represent an aggregated systemthat manages more than one DER.

In the example of FIG. 1, each of control devices 10 may receivecoefficient values 14. Control devices 10 may use coefficient values 14to determine one or more power setpoint values that dictate the levelsof real and/or reactive power that are to be outputted by the associatedenergy resource or energy resources. For example, control device 10Awould determine setpoint values for energy resource 8A and controldevice 10B would determine respective setpoint values for energyresources 8B, 8C, and 8D. Each of control devices 10 may determinesetpoint values based on the coefficient values 14 and previous setpointvalues for the control device. Determination of setpoint values isfurther described with respect to FIGS. 2-5 below. Control devices 10may use the determined setpoint values to manage the power outputs ofenergy resources 8.

Energy resources 8 may, in various examples, represent any device orsystem capable of generating electrical power. In the example of FIG. 1,for instance, energy resources 8A and 8B are PV panels, energy resource8C is an energy storage device (e.g., a battery or battery system), andenergy resource 8D is an EV. Other examples of energy resources includegenerators (e.g., gas generators, etc.), fuel cells, and others.

While certain operations are described in the example of FIG. 1 as beingperformed by power management unit 4 or control devices 10, theseoperations may, in other examples, be performed by one or more othercomponents of system 2, or by components not shown in FIG. 1. In someexamples, for instance, each of control devices 10 may be configured toreceive measurements 12 and power measurement 13 and determinecoefficient values 14. As another example, power management unit 4 may,in some instances, be configured to determine setpoint values for eachof control devices 10. In yet another example, one or both of theseoperations may be performed by one or more standalone computing devices(not shown) that communicate with control devices 10. This may be thecase when, for example, one or more of control devices 10 are legacydevices that do not have sufficient computing power or communicationsabilities.

By iteratively determining power setpoints on a real-time or nearreal-time basis, performance of system 2 may achieve near optimummanagement of the power system without requiring complex orcomputationally powerful components. Additionally, by incorporatingvoltage measurements, current measurements, and power measurements, thetechniques described herein ensure that limits on these quantities arenot violated.

The mathematical development of the control techniques described hereinis detailed below. Upper-case and lower-case boldface letters will beused for matrices and column vectors; (·)^(T) for transposition; (·)*for complex-conjugate; and (·)^(H) for complex-conjugate transposition.

{·} and

{·} denote the real and imaginary parts of a complex number,respectively. j:=√{square root over (−1)}, the imaginary unit; and |·|denotes the absolute value of a number or the cardinality of a(discrete) set. For a given N×1 vector x ε

^(N), |x| takes the absolute value entry-wise; ∥x∥₂:=√{square root over(x^(H)x)}; and diag(x) returns a N×N matrix with the elements of x inits diagonal. Given a given matrix X ε

^(N×M), x_(m,n) denotes its (m, n)-th entry and ∥X˜₂ denotes thel₂-induced matrix norm. For a function ƒ:

^(N)→

, ∇_(x)ƒ(x) returns the gradient vector of ƒ(x) with respect to x εR^(N). 1_(N) denotes the N×1 vector with all ones, and 0_(N) denotes theN×1 vector with all zeros. Given two sets χ₁ ⊂

^(N) and χ₂ ⊂

^(N), χ₁ ⊕ χ₂ denotes the Minkowski sum of χ₁ and χ₂. Finally,proj_(χ){x} denotes the projection of x onto the convex set χ.

The present disclosure addresses two classes of DERs: i) devices thatare individually controllable; and, ii) groups of DERs that can becontrolled as a whole. The second class may model, for example,residential homes and buildings with multiple DERs behind the meter,renewable-based systems with multiple inverters or microinverters,parking garages for EVs, and other aggregate systems. Each DER may beeither wye-connected or delta-connected to the power system, and it canbe either single-phase or three-phase. In the following, pertinentnotation and modeling details are outlined.

For future developments, let

:={a, b, c} ∪ {ab, bc, ca} be the set of possible connections, with {a,b, c} referring to wye connections (e.g., line to ground) and {ab, bc,ca} referring to delta connections (e.g., line to line).

Let

:={1, . . . , D} be the set of individually-controllable DERs, and letx_(j):=[P_(j),Q_(j)]^(T) ε

² collect the real and reactive power setpoint of DER j ε

. The DER can be either wye-connected or delta-connected to the network.Three-phase DERs are assumed to operate in a balanced setting; thus, thesetpoint x_(j) is the same across phases. The set

_(j) ⊂

collects the phases where DER j is connected.

Denote as χ_(j) ⊂

² the set of possible power setpoints x_(j) for the DER j. The set χ_(j)captures hardware and operational constraints and it is assumed to beconvex and compact. It is assumed that the DERs are endowed with controldevices (e.g., control devices 10 of FIG. 1) that are designed so that,upon receiving the setpoint x_(j) ε χ_(j), the output powers of the DERsare driven to the commanded setpoints. Relevant dynamical models for theoutput powers of an inverter-interfaced DER are discussed in the art andcan be found in datasheets of commercially available DERs.

For an inverter-interfaced DER, consider the following prototypicalrepresentation of the set χ_(j):

χ_(j)( p,p,r):={[P _(j) ,Q _(j)]^(T) :p≤P _(j)≤ p,P _(j) ² +Q _(j) ² ≤r²}  (1)

where p, p, and r>0 are given DER-dependent parameters. For example, fora PV system, r represents the inverter capacity, p=0, and p is theavailable real power. For an energy storage system, r represents theinverter rating, and p; p are updated during the operation of thebattery based on the current state of charge. Notice that the set χ_(j)is typically time varying, as the parameters p, p, and r vary over timebased on ambient conditions and/or internal DER state.

On the other hand, consider the following operating region for DERs withcontrollable active powers (e.g., variable speed drives, EVs, etc.):

χ_(j)( p,p ):={[P _(j) ,Q _(j)]^(T) :p≤P _(j) ≤p,Q _(j)=0}.  (2)

With regard to DERs having discrete controls, let {tilde over (χ)}_(j) ⊂

² denote the nonconvex operating region of a DER with a discrete set ofimplementable power setpoints. This may be the case, for example, forHVAC systems where {tilde over (χ)}_(j)={[P_(j),Q_(j)]^(T):P_(j) ε{0,p}, Q_(j)=0}, or EVs with discrete charging levels. For thesedevices, the set χ_(j) is the convex hull of {tilde over (χ)}_(j); i.e.,χ_(j):=ch {tilde over (χ)}_(j). For example, for an HVAC system,χ_(j)={[P_(j),Q_(j)]^(T):0≤P_(j)≤p, Q_(j)=0}. The control techniquesdescribed herein utilize a randomization procedure to recoverimplementable setpoints based on χ_(j). For a DER with discretecontrols, {tilde over (x)}_(j) ε {tilde over (χ)}_(j) denotes animplementable setpoint, whereas x_(j) Π χ_(j) is a (relaxed) setpointcomputed based on the convex hull of χ_(j).

In some examples, the power system may feature a set

:={1, . . . , D} of residential homes, building, or other facilitiesthat have multiple DERs that are controlled jointly. Let

_(j):={1, . . . , D _(j)} denote the set of devices within the jthaggregation, and define as x _(j):=Σ_(iεD) _(j) x_(i) the setpoint forthe net powers generated by the DERs within a group. The set

_(j) ⊂

collects the connections of the aggregation j.

Let χ _(j) ⊆ ⊕_(iεD) _(j) χ_(j) be either the exact Minkowski sum of theoperating regions of the DERs within the jth aggregation or anappropriate inner approximation. Notice that if a DER i involvesdiscrete controls, the convex hull χ_(j)=ch {tilde over (χ)}_(j) may beutilized to compute the (inner approximation of) the Minkowski sum. Inthe following, pertinent results for the Minkowski sum of sets (1) and(2) are provided. First, notice that the Minkowski sum of two setsχ_(j)(p _(j),p _(j)) and χ_(n)(p _(n),p _(n)) for two DERs withcontrollable active powers is given by:

χ_(j)( p _(j) ,p _(j))⊕χ_(n)( p _(n) ,p _(n))={ x=[P,Q] ^(T) :p _(j) +p_(n) ≤P≤p _(j) +p _(n) , Q=0}.  (3)

The following theorems will deal with the Minkowski sum of the sets

χ_(j)( p _(j) ,p _(j) ,r _(j))⊕χ_(n)( p _(n) ,p _(n)) and χ_(j)( p _(j),p _(j) ,r _(j))⊕χ_(n)( p _(n) ,p _(n) ,r _(n)).

The Minkowski sum between χ(p ₁,p ₁,r₁) and χ(p ₂,p ₂,r₂) in (1) and(2), respectively, is given by

$\begin{matrix}{{{\chi\left( {{\underset{\_}{p}}_{1},{\overset{\_}{p}}_{1},r_{1}} \right)} \oplus {\chi\left( {{\underset{\_}{p}}_{2},{\overset{\_}{p}}_{2},r_{2}} \right)}} = \left\{ {{{{\left\lbrack {P,Q} \right\rbrack^{T}\text{:}{\underset{\_}{p}}_{1}} + {\underset{\_}{p}}_{2}} \leq P \leq {{\overset{\_}{p}}_{1} + {\overset{\_}{p}}_{2}}},{{Q} \leq r},{{\left( {{\underset{\_}{p}}_{2} - P} \right)^{2} + Q^{2}} \leq r^{2}},{{\left( {P - {\overset{\_}{p}}_{2}} \right)^{2} + Q^{2}} \leq r^{2}}} \right\}} & (4)\end{matrix}$

and it is convex and compact. This is hereinafter referred to as Theorem1.

Inner and outer approximations of the Minkowski sum of two sets χ(p ₁,p₁,r₁) and χ(p ₂,p ₂,r₂) are given by

χ( p ₁ +p ₂ ,p ₁ +p ₂,ρ⊆χ( p ₁ ,p ₁ ,r ₁)⊕χ( p ₂ ,p ₂ ,r ₂)  (5a)

⊆χ( p ₁ +p ₂ ,p ₁ +p ₂ ,r ₁ +r ₂)  (5b)

for any ρ>0 satisfying the following condition

ρ² ≤r ₁ ² +r ₂ ²+α−β₁−β₂+2√{square root over ((r ₁ ²−β₁)(r ₂²−β₂))}  (6)

where α:=[max{p ₁+p ₂, min{0,p ₁+p ₂,}}]², and β_(i):=max{p _(i) ²,p_(i) ²}, i=1,2. This is hereinafter referred to as Theorem 2.

Notice that the inner approximation χ(p ₁+p ₂,p ₁+p ₂,ρ) is convex andcompact, and it is in the form of (1). Expression (3) along with theresults of Theorem 1 and Theorem 2 can be utilized to compute an innerapproximation of the feasible region of the net powers x _(j) for eachaggregation of DERs j ε

. For example, the feasible region for the net powers generated by aresidential house with a PV system, a battery, and an EV can be computedby first leveraging (5a) to sum the sets pertaining to the PV system andthe battery and subsequently (4).

Next, consider a generic multi-phase power system (e.g., a powerdistribution network) with multiphase nodes collected in the set

∪ {0},

:={1, . . . , N}, and power lines (e.g., distribution lines) representedby the set of edges ϵ:={(m, n)}⊂ (

∪ {0})×(

∪ {0}). Node 0 denotes the three-phase slack bus (e.g., the point ofconnection of the power system with the rest of the electrical system).At each multiphase node, controllable and non-controllable devices canbe either wye- or delta-connected.

Presented below is a brief showcase of the set of AC power-flowequations for this generic setting. Let v be a vector collecting theline-to-ground voltages in all phases of the nodes in

; similarly, vector i collects all the phase net current injections,i^(Δ) the phase-to-phase currents in all the delta connections, andvectors s^(Y) and s^(Δ) collect the net complex powers injected at nodes

from devices with wye and delta connections, respectively. With thesedefinitions in place, the AC power-flow equations can be compactlywritten as:

diag(H ^(T)(i ^(Δ))*)v+s ^(Y)=diag(v)i*,  (7a)

s ^(Δ)=diag(Hv)(i ^(Δ))*,  (7b)

i=Y _(L0) v0+Y _(LL) v,  (7c)

where Y₀₀ ε

^(3×3), Y_(L0) ε

^(N) ^(ϕ) ^(×3), Y_(0L) ε

^(3×N) ^(ϕ) , and Y_(LL) ε

^(N) ^(ϕ) ^(×N) ^(ϕ) are the submatrices of the admittance matrix

$\begin{matrix}{{Y:={\begin{bmatrix}Y_{00} & Y_{0L} \\Y_{L\; 0} & Y_{LL}\end{bmatrix} \in {\mathbb{C}}^{N_{\varphi} + {3 \times N_{\varphi}} + 3}}},} & (8)\end{matrix}$

which can be formed from the topology of the network and the π-model ofthe distribution lines. N_(ϕ) is the total number of single-phaseconnections and H is a N_(ϕ)×N_(ϕ) block-diagonal matrix mapping thedirection of the currents i^(Δ) in the delta connections.

The nonlinearities in (7) hinder the possibility of seeking analyticalclosed-form solutions to pertinent electrical quantities such asvoltages, power flows, and line currents as a function of the DERs'power injections. To facilitate the design and analysis of real-timeoptimization methods, the techniques described herein leverageapproximate linear models developed in the literature. To this end,denote as

the vector collecting the phase-to-ground voltages at given measurementpoints;

the vector collecting the line currents for a subset of monitoreddistribution lines (or given by pseudo-measurements); and, p₀ ε

³ the vector of real powers entering node 0 on the phases {a, b,c}.Then, |

| (where the absolute value is taken entry-wise), |

|, and p₀ can be approximately expressed as:

|

(x,x )|=

A _(j) x _(j)+

Ā _(j) x _(j) +a  (9a)

a:=

A _(j,ϕ) l _(j,ϕ) +a ₀  (9b)

|

(x,x )|=

B _(j) x _(j)+

B _(j) x _(j) +b  (10a)

b:=

B _(j,ϕ) l _(j,ϕ) +b ₀  (10b)

{tilde over (p)} ₀(x,{tilde over (x)})=

M _(j) x _(j) +

M _(j) x _(j) +m  (11a)

m:=

M _(j,ϕ) l _(j,ϕ) +m ₀  (11b)

where l_(j,ϕ) ε

² collects the net non-controllable active and reactive powers atconnection ϕ ε

of node n ε

, x and x stack all the setpoints {x_(j)} and x _(j), respectively, andthe matrices A_(j,ϕ), Ā_(j,ϕ), B_(j,ϕ), B _(j,ϕ), M_(j,ϕ), M _(j,ϕ)along with the vectors a₀, b₀, and m₀ are model parameters that can becomputed through e.g., the fixed-point linearization method. Forbrevity, define the matrices A_(j):=

A_(j,ϕ), Ā_(j):=

Ā_(j,ϕ), B_(j):=

B_(j,ϕ), B _(j):=

B _(j,ϕ), M_(j):=

M_(j,ϕ), M _(j):=

M _(j,ϕ). These model parameters capture the effects of different typesof connection (e.g., wye or delta) and can be computed based on theadmittance matrix of the system. If a fixed-point linearization methodis utilized, knowledge of the non-controllable powers l_(j,ϕ) is notrequired for the computation of the model parameters. If only wyeconnections are present, alternative ways to obtain (9)-(11) may beused.

It is worth emphasizing that the approximate models (9)-(11) areutilized to facilitate the design and the performance analysis of thereal-time algorithm. However, appropriate measurements from distributiongrid and DERs may be accommodated as described herein to cope withinaccuracies in the representation of the AC power flows and stabilityclaims under the (realistic) nonlinear model are established below.

Hereafter, the subscripts

_(v) and

_(i) will be dropped from (9) and (10) for notational simplicity, withthe understanding that functions v(x, x) and i_(L)(x, x) refer tovoltages and currents at given points of interest.

From this setup, the real-time optimization techniques may be designedsuch that power setpoints of the DERs are updated on a second-wise (orsimilar) timescale to maximize operational objectives while coping withthe variability of ambient conditions and non-controllable assets.Consider, then, discretizing the temporal domain as t_(k)=kh, where k ε

and h>0 will be taken to be the time required to compute one closed-loopiteration of the process disclosed herein. As discussed shortly, thevalue of h is based on underlying communication delays, as well asoperational considerations of utility and aggregators.

Next, the time-varying optimization formalism may be leveraged to modeloptimal operational trajectories for the DERs, based on 1) possiblytime-varying optimization objectives, engineering limits (e.g., voltagelimits and Ampacity limits), as well as 2) variability ofnon-controllable assets and ambient conditions. Hereafter, thesuperscript ^((k)) will be utilized to indicate variables, functions,and inputs at time t_(k), for all k ε

.

Let ν^(min) and ν^(max) be given limits for the magnitude ofphase-to-ground voltages (e.g., ANSI C.84.1 limits), and let i^(max) bea vector collecting the Ampacity limits for the monitored distributionlines. Finally, s^((k)) ε {0,1} indicates whether the power system isrequested to follow a setpoint p_(0,set) ^((k)) for the real powers atthe three phases of the point of connection with the rest of theelectrical network. When s^((k))=1, the sequence of setpoints {p_(0,set)^((k))}_(k) shall be tracked within a given accuracy E^((k)). With thesedefinition, the following time-varying optimization problem isformulated to model optimal operational trajectories {x_(j) ^(opt), k ε

} for the DERs:

$\begin{matrix}{{\left( {P\; 1^{(k)}} \right){\min\limits_{x,\overset{\_}{x}}{\sum_{j \in }{f_{j}^{(k)}\left( x_{j} \right)}}}} + {\sum_{j \in \overset{\_}{}}{{\overset{\_}{f}}_{j}^{(k)}\left( {\overset{\_}{x}}_{j} \right)}}} & \left( {12a} \right) \\{{{subject}\mspace{14mu} {to}\text{:}\mspace{14mu} x_{j}} \in {\chi_{j}^{(k)}{\forall{j \in }}}} & \left( {12b} \right) \\{{\overset{\_}{x}}_{j} \in {{\overset{\_}{\chi}}_{j}^{(k)}{\forall{j \in \overset{\_}{}}}}} & \left( {12c} \right) \\{{s^{(k)}{I_{3}\left( {{{\overset{\sim}{p}}_{0}^{(k)}\left( {x,\overset{\_}{x}} \right)} - p_{0,{set}}^{(k)}} \right)}} \leq {E^{(k)}1_{3}}} & \left( {12d} \right) \\{{s^{(k)}{I_{3}\left( {p_{0,{set}}^{(k)} - {{\overset{\sim}{p}}_{0}^{(k)}\left( {x,\overset{\_}{x}} \right)}} \right)}} \leq {E^{(k)}1_{3}}} & \left( {12e} \right) \\{{{{\overset{\sim}{v}}^{(k)}\left( {x,\overset{\_}{x}} \right)}} \leq {v^{{ma}\; x}1}} & \left( {12f} \right) \\{{v^{m\; i\; n}1} \leq {{{\overset{\sim}{v}}^{(k)}\left( {x,\overset{\_}{x}} \right)}}} & \left( {12g} \right) \\{{{{\overset{\sim}{\iota}}^{(k)}\left( {x,\overset{\_}{x}} \right)}} \leq i^{{ma}\; x}} & \left( {12h} \right)\end{matrix}$

recalling that χ_(j) ^((k)) is a convex set modeling hardwareconstraints of the DER j at a given time τ_(k), ƒ_(j) ^((k)):

²→

is a time-varying convex function associated with the DER j ε

, and the function ƒ _(j) ^((k)):

²→

associated with the jth aggregation of DERs is defined as follows:

$\begin{matrix}{{{\overset{\_}{f}}_{j}^{(k)}\left( x_{j} \right)} = {\min\limits_{{{\{ x_{i}\}}i} \in _{j}}{\sum_{i \in {\overset{\_}{}}_{j}}{f_{i}^{(k)}\left( x_{i} \right)}}}} & \left( {13a} \right) \\{{{{subject}\mspace{14mu} {to}\text{:}\mspace{14mu} x_{i}} \in \chi_{i}^{(k)}},{\forall{i \in {\overset{\_}{}}_{j}}}} & \left( {13b} \right) \\{{\sum_{i \in {\overset{\_}{}}_{j}}x_{i}} = {{\overset{\_}{x}}_{j}.}} & \left( {13c} \right)\end{matrix}$

Problem (13) is utilized to disaggregate the set setpoint x _(j) acrossthe DERs i ε

_(j).

Before proceeding, it is worth noting the following for the bi-levelformulation (12)-(13):

-   -   i) when set χ _(j) ^((k)) is given by the (exact) Minkowski sum        of χ_(i) ^((k)), i ε        _(j), (12)-(13) is equivalent to a “flat” optimization strategy        where (12) does not consider points of aggregation (thus, the        flat formulation includes individual optimization variables and        constraints that pertain to all the DERs);    -   ii) if the set χ _(j) ^((k)) is an inner approximation of the        Minkowski sum, then (12)-(13) represents a restriction of the        “flat” optimization problem.

Problem (P1^((k))) is a time-varying convex optimization problem;however, solving (P1^((k))) in a batch fashion at each time τ_(k) mightbe impractical, possibly because of the following three challenges:

-   -   c1: Complexity. For real-time implementations (e.g., when h is        on the order of a second or a few seconds), it might be        unfeasible to solve (P1 ^((k))) to convergence; this is        especially the case of distributed settings, where multiple        communication rounds are required to reach convergence.    -   c2. Model inaccuracy. The linear models (9)-(11) provide only an        approximate relationship between power injections. As a result,        the optimal solution of (P1 ^((k))) might not necessarily be        feasible for the actual system.    -   c3. Pervasive metering. Solving (P1 ^((k))) (either in a batch        form or online) requires collecting measurements of the        (aggregate) noncontrollable loads l_(j,Φ) at all locations in        real time.

The techniques described herein use the following feedback-based onlinealgorithm that tracks the optimal solution of (P1 ^((k))) over time,while coping with model inaccuracies and avoiding ubiquitous metering.

The following assumption is imposed throughout. For each DER i ε

, and for each DER i ε

_(j) in the aggregation j ε

:

-   -   i the set χ_(i) ^((k)) is convex and compact for all τ_(k);    -   ii function ƒ_(i) ^((k)) (x_(i)) is convex and continuously        differentiable, and its gradient is    -   Lipschitz continuous for all τ_(k).        This is hereinafter referred to as Assumption 1.

Start by outlining the results pertaining to the DER aggregations

. Suppose that problem (13) is feasible and Assumption 1 holds. Then,the dual function associated with problem (13) is strongly concave.Moreover, the unique optimal dual variable associated with (13c) isbounded.

The function ƒ _(j) ^((k)) (x) is convex. Under Assumption 1, thefunction ƒ _(j) ^((k)) (x) is Lipschitz continuous. Further, thegradient of ƒ _(j) ^((k)) (x) evaluated at x _(j) ^((k)) is given by:

$\begin{matrix}{\left. {\nabla_{\overset{\_}{x}}{\overset{\_}{f}}_{j}^{(k)}} \right|_{\overset{\_}{x} = {\overset{\_}{x}}_{j}^{(k)}} = {- \xi_{j}^{(k)}}} & (14)\end{matrix}$

where ξ_(j) ^((k)) is the optimal dual variable associated withconstraint (13c). This is hereinafter referred to as Theorem 3.

Under Assumption 1, the gradient ∇ _(x) ƒ _(j) ^((k)) is Lipschitzcontinuous. This is hereinafter referred to as Theorem 4.

Notice that Assumption 1.ii can be relaxed. In fact, if the functionƒ_(i) ^((k)) (x_(i)) is not strongly convex, one can regularize theproblem (13) with a strongly convex term; e.g., the cost can beregularized as

_(j) ƒ_(i) ^((k)) (x_(j))+r∥x_(i)∥₂ ², for a given constant r>0. Theresults of Theorem 3 and Theorem 4 are valid at each time instant τ_(k).These results are utilized for the design of the real-time algorithm toupdate the aggregate setpoint x ^((k)), and to establish pertinentconvergence and stability claims.

Let λ^((k)), μ^((k)), γ^((k)), ν_((k)), and ξ^((k)) be the dualvariables associated with constraints (12d), (12e), (12f), (12g), and(12h), respectively. The Lagrangian function associated with the problem(12) at time τ_(k) is given by:

$\begin{matrix}{{L^{(k)}\left( {x,\overset{\_}{x},d} \right)}:={{\sum\limits_{i \in }{f_{j}^{(k)}\left( x_{j} \right)}} + {\sum\limits_{j \in \overset{\_}{}}{{\overset{\_}{f}}_{j}^{(k)}\left( {\overset{\_}{x}}_{j} \right)}} + {\sum\limits_{j \in }\left\lbrack {{{s^{(k)}\left( {\lambda - v} \right)}^{T}M_{j}x_{j}} + {\left( {\gamma - \mu} \right)^{T}A_{j}x_{j}} + {\zeta^{T}B_{j}x_{j}}} \right\rbrack} + {\sum\limits_{j \in \overset{\_}{}}\left\lbrack {{{s^{(k)}\left( {\lambda - v} \right)}^{T}{\overset{\_}{M}}_{j}x_{j}} + {\left( {\gamma - \mu} \right)^{T}{\overset{\_}{A}}_{j}{\overset{\_}{x}}_{j}} + {\zeta^{T}{\overset{\_}{B}}_{j}{\overset{\_}{x}}_{j}}} \right\rbrack} + {{s^{(k)}\left( {\lambda - v} \right)}^{T}\left( {m_{0} - p_{0,{set}}^{(k)}} \right)} - {\left( {\lambda - v} \right)^{T}E^{(k)}1} + {\gamma^{T}\left( {a_{0}^{(k)} - {v^{{ma}\; x}1}} \right)} + {\mu^{T}\left( {{v^{m\; i\; n}1} - a_{0}^{(k)}} \right)} - {\zeta^{T}i^{{ma}\; x}}}} & (15)\end{matrix}$

where d:=[γ^(T),ν^(T),λ^(T),μ^(T),ξ^(T)]^(T) for simplicity ofexposition and 1 is a vector of ones of appropriate dimensions. Considerthe following regularized Lagrangian function, where r_(p),r_(d)>0 areregularization factors:

$\begin{matrix}{{L_{r}^{(k)}\left( {x,\overset{\_}{x},d} \right)}:={{L^{(k)}\left( {x,\overset{\_}{x},d} \right)} + {\frac{r_{p}}{2}{x}_{2}^{2}} + {\frac{r_{p}}{2}{\overset{\_}{x}}_{2}^{2}} - {\frac{r_{d}}{2}{d}_{2}^{2}}}} & (16)\end{matrix}$

and notice that L_(r) ^((k)) (x, x, d) is strongly convex in the primalvariables and strongly concave in the dual variables. Consider then thefollowing time-varying saddle-point problem:

$\begin{matrix}{\max\limits_{d \in {\mathbb{R}}_{+}^{{2{\mathcal{M}_{v}}} + {\mathcal{M}_{i}} + 3}}{\min\limits_{{\{ x_{j}\}},{\{{\overset{\_}{x}}_{j}\}}}{L_{r}^{(k)}\left( {x,\overset{\_}{x},d} \right)}}} & (17)\end{matrix}$

and let z^((k,*))=|[(x^((k,*)))^(T)·(x^((k,*)))^(T)·(d^((k,*)))^(T)]^(T): unique primal-dual optimizer of(17). The design of the online algorithm leverages appropriatemodifications of online projected-gradient methods to track thetime-varying solution of (17). Although the optimizer of (17) isexpected to be different from optimizers of the original problem

${\max\limits_{d \in {\mathbb{R}}_{+}^{{2{\mathcal{M}_{v}}} + {\mathcal{M}_{i}} + 3}}{\min\limits_{{\{ x_{j}\}},{\{{\overset{\_}{x}}_{j}\}}}{L_{r}^{(k)}\left( {x,\overset{\_}{x},d} \right)}}},$

shown below that the strong convexity and concavity of L_(r)^((k))(x,x,d) allows the real-time algorithm to achieve Q-linearconvergence. The discrepancy between x^((k,*)),x ^((k,*)) and thesolution of problem (P1 ^((k))) can be bounded using known techniques.

Let α>0 be a given step size. Then, given the results of Theorem 3 andbased on the regularized time-varying saddle-point formulation (17), theexecution of the following operations at each time t_(k) defines themethod for real-time optimization of the power system. The methodproduces power setpoints for the DERs at each time t_(k), k ε

.

At each t_(k) perform the following operations:

[S1 a]: Collect voltage measurements |{circumflex over (v)}^((tk))| atgiven measurement points

, and perform the following updates:

$\begin{matrix}{\gamma^{({k + 1})} = {{proj}_{{\mathbb{R}}_{+}^{\mathcal{M}_{v}}}\left\{ {\gamma^{(k)} + {\alpha \left( {{v^{m\; i\; n}1} - {{\hat{v}}^{({tk})}} - {r_{d}\gamma^{(k)}}} \right)}} \right\}}} & (18) \\{\mu^{({k + 1})} = {{proj}_{{\mathbb{R}}_{+}^{\mathcal{M}_{v}}}\left\{ {\mu^{(k)} + {\alpha \left( {{{\hat{v}}^{({tk})}} - {v^{{ma}\; x}1} - {r_{d}\mu^{(k)}}} \right)}} \right\}}} & (19)\end{matrix}$

[S1 b]: Obtain measurements or estimates of î_(L) ^((k)) on lines ofinterest and perform the following updates:

$\begin{matrix}{\zeta^{({k + 1})} = {{proj}_{{\mathbb{R}}_{+ \;}^{\mathcal{M}_{i}}}\left\{ {\zeta^{(k)} + {\alpha \left( {{{\hat{\iota}}_{L}^{(k)}} - i^{{ma}\; x} - {r_{d}\zeta^{(k)}}} \right)}} \right\}}} & (20)\end{matrix}$

[S1 c]: Collect measurements at point of coupling and perform thefollowing updates:

$\begin{matrix}{\lambda^{({k + 1})} = {{proj}_{{\mathbb{R}}_{+}^{3}}\left\{ {\lambda^{(k)} + {\alpha \left( {{\hat{p}}_{0}^{(k)} - p_{0,{set}}^{(k)} - {E^{(k)}1_{3}} - {r_{d}\lambda^{(k)}}} \right)}} \right\}}} & (21) \\{v^{({k + 1})} = {{proj}_{{\mathbb{R}}_{+}^{3}}\left\{ {v^{(k)} + {\alpha \left( {p_{0,{set}}^{(k)} - {\hat{p}}_{0}^{(k)} - {E^{(k)}1_{3}} - {r_{d}v^{(k)}}} \right)}} \right\}}} & (22)\end{matrix}$

[S2 a]: Each device j ε

performs the following operations:

[S2 a.1] Measure output powers {circumflex over (x)}_(j) ^((k))

[S2 a.2] Update power setpoints x_(j) ^((k+1)) as follows:

$\begin{matrix}{x_{j}^{({k + 1})} = {{proj}_{\chi^{(k)}}\left\{ {{\hat{x}}_{j}^{(k)} - {\alpha \left( {{\nabla_{x_{j}}{f_{j}^{(k)}\left( {\hat{x}}_{j}^{(k)} \right)}} + {{s^{(k)}\left( {\lambda^{({k + 1})} - v^{({k + 1})}} \right)}^{T}M_{j}} + {\zeta^{({k + 1})}B_{j}} + {\left( {\gamma^{({k + 1})} - \mu^{({k + 1})}} \right)^{T}A_{j}} + {r_{p}{\hat{x}}_{j}^{(k)}}} \right)}} \right\}}} & (23)\end{matrix}$

[S2 a.3] If DER j ε

has a set of discrete setpoints, determine the implementable setpointas:

$\begin{matrix}{e_{j}^{(k)} = {\sum\limits_{ = 1}^{k}\left( {x_{j}^{()} - {\overset{\sim}{x}}_{j}^{()}} \right)}} & (24) \\{{\overset{\sim}{x}}_{j}^{({k + 1})} \in {{proj}_{{\overset{\sim}{x}}_{j}^{(k)}}{\left\{ {x_{j}^{({k + 1})} + e_{j}^{(k)}} \right\}.}}} & (25)\end{matrix}$

[S2 a.4] Command setpoint to the DER.

[S2 b]: Each DER aggregation j ε

performs the following operations:

[S2 b.1] Measure aggregate output powers

[S2 a.2] Update setpoints for the aggregate powers x _(j) ^((k+1)) asfollows:

$\begin{matrix}{{\overset{\_}{x}}_{j}^{({k + 1})} = {{proj}_{{\overset{\_}{\chi}}^{(k)}}\left\{ {{\overset{\overset{}{\_}}{x}}_{j}^{(k)} - {\alpha \left( {{- \xi_{j}^{(k)}} + {{s^{(k)}\left( {\lambda^{({k + 1})} - v^{({k + 1})}} \right)}^{T}{\overset{\_}{M}}_{j}} + {\zeta^{({k + 1})}{\overset{\_}{B}}_{j}} + {\left( {\gamma^{({k + 1})} - \mu^{({k + 1})}} \right)^{T}{\overset{\_}{A}}_{j}} + {r_{p}{\overset{\overset{}{\_}}{x}}_{j\;}^{(k)}}} \right)}} \right\}}} & (26)\end{matrix}$

[S2 b.3] Given the aggregate powers x _(j) ^((k+1)), determine thesetpoints

of the individual DERs

_(j) and the new vector ξ_(j) ^((k+1)) by solving the saddle-pointproblem:

$\begin{matrix}{{\max\limits_{\xi}{\min\limits_{{\{{x_{i} \in \chi_{i}^{(k)}}\}}_{i \in {\overset{\_}{}}_{j}}}{\sum_{i \in {\overset{\_}{}}_{j\;}}{f_{i}^{(k)}\left( x_{i} \right)}}}} + {\xi^{T}\left( {{\sum_{i \in {\overset{\_}{}}_{j}}x_{i}} - {\overset{\_}{x}}_{j}^{({k + 1})}} \right)}} & (27)\end{matrix}$

[S2 b.4] if DER j ε

_(j) has a set of discrete setpoints, determine the implementablesetpoint as:

$\begin{matrix}{e_{j}^{(k)} = {\sum\limits_{ = 1}^{k}\left( {x_{j}^{()} - {\overset{\sim}{x}}_{j}^{()}} \right)}} & (28) \\{{\overset{\sim}{x}}_{j}^{({k + 1})} \in {\arg \; {\min_{x \in {\overset{\sim}{\chi}}_{j}^{(k)}}{{{x - \left( {x_{j}^{({k + 1})} + e_{j}^{(k)}} \right)}}_{2}.}}}} & (29)\end{matrix}$

[S2 a.5] Command setpoint to the DER.

-   -   [S3]: Go to [S1].

FIG. 2 is a conceptual diagram illustrating another example powermanagement system configured to manage energy resources in a powersystem, in accordance with one or more aspects of the presentdisclosure. Specifically, the power management system shown in FIG. 2implements the control method described by operations [S1]-[S3]. Thisreal-time method affords a distributed implementation as shown in FIG.2. Once measurements {circumflex over (v)}^((k)), î_(L) ^((k)), and{circumflex over (p)}₀ ^((k)) are acquired, operation [S1] may beperformed at the utility/aggregator, which subsequently broadcasts theupdated dual variables. Operations [S2 a] and [S2 b] may be implementedlocally at individual DERs and aggregations of DERS, respectively.

Finally, notice that operations (25) and (29), which are performed forDERs with discrete setpoints, involve the solution of a localizednonconvex program to compute implementable commands.

The ability of the techniques described herein to track the optimizersz^((k,+)) of (17) is analytically established below.

start by stating the following assumption regarding problem (12).Problem (12) is feasible and Slater's condition holds at each timet_(k), k ε

. This is hereinafter referred to as Assumption 2.

Assumption 2 implies that there exists a power flow solution thatadheres to voltage and ampacity limits. When the power system isrequired to follow a setpoint at the point of common coupling,Assumption 2 presumes that the setpoint is feasible. Feasibility of thepower flow solutions (with and without setpoints for the active andreactive power at the substation) can be assessed by solving suitableoptimization problems at a slower time scale.

Regarding the temporal variability of problem (12), the followingquantity is introduced to capture the variation of the optimal solutiontrajectory over time:

σ^((k)):=∥z^((k+1,*))−z^((k,*)∥) ₂≤σ  (30)

for a given σ>0. For sufficiently small sampling intervals h, σ can beinterpreted as a bound on the norm of the gradient of the optimalsolution trajectory

In the context of (12), σ depends on the variability of the costfunction, non-controllable loads, as well as available powers from therenewable-based DERs.

Next, since models (9)(11) are linear and the sets {χ_(j) ^((k))} and {χ_(j) ^((k))} are compact, there exist constants G_(ν)≤+∞, G₀≤+∞, andG_(L)≤+∞ such that

∥∇_([x,x]) {tilde over (v)} ^((k))(x,x )∥₂ ≤G _(ν),

∥∇_([x,x]) {tilde over (p)} ₀ ^((k))(x,x )∥₂ ≤G ₀,

∥∇_([x,x]) {tilde over (e)} _(L) ^((k))(x,x )∥₂ ≤G _(L).

For future developments, define G :=max{G_(ν),G₀,G_(L)}. Further, noticethat from Assumption 1 and Theorem 4, the gradient mapg^((k))(x,x):=[∇_(x) ₁ ^(T)ƒ₁ ^((k))(x₁), . . . ,

), ∇ _(x) ₁ ^(T) ƒ ₁ ^((k))(x ₁), . . . ,

)]^(T) is Lipschitz continuous with a given constant L^((k)) over theset χ^((k)):=χ₁ ^((k))× . . . ×

×χ ₁ ^((k))× . . . ×

. Let L:=lim sup{L^((k))}, so that

∥g ^((k))(x,x )−g ^((k))(x′,x′)∥₂≤L∥x−x′∥ ₂  (31)

for all x, x′ ε χ^((k)) and t_(k), k ε

.

Define the errors introduced by measurement noise and modelingmismatches (i.e., discrepancy between the nonlinear AC power-flowequations and the linearized model, as well as possible inaccurateknowledge of the admittance matrix) as follows:

$\begin{matrix}{e_{x}^{(k)}:={{\begin{bmatrix}x^{(k)} \\{\overset{\_}{x}}^{(k)}\end{bmatrix} - \begin{bmatrix}{\hat{x}}^{(k)} \\{\overset{\overset{}{\_}}{x}}^{(k)}\end{bmatrix}}}_{2}} & (32) \\{e_{0}^{(k)}:={{{{\overset{\sim}{p}}_{0}\left( {x^{(k)},{\overset{\_}{x}}^{(k)}} \right)} - {\hat{p}}_{0}^{(k)}}}_{2}} & (33) \\{e_{v}^{(k)}:={{{{\overset{\sim}{v}}^{(k)}\left( {x^{(k)},{\overset{\_}{x}}^{(k)}} \right)} - {{\hat{v}}^{(k)}}}}_{2}} & (34) \\{e_{L}^{(k)}:={{{{\overset{\sim}{\iota}}_{L}^{(k)}\left( {x^{(k)},{\overset{\_}{x}}^{(k)}} \right)} - {{\hat{\iota}}_{L}^{(k)}}}}_{2}} & (35)\end{matrix}$

recalling that {circumflex over (v)}^((k)), î_(L) ^((k)), and{circumflex over (p)}₀ ^((k)) are actual measurements (orpseudo-measurements). The following assumption is made. There existfinite constants e_(x), e₀, e_(ν), and e_(L) such that e_(x)^((k))≤e_(x), e₀ ^((k))≤e₀, e_(ν) ^((k))≤e_(ν), and e_(L) ^((k))≤e_(L)for all t_(k); that is, the errors (32)-(35) are uniformly bounded intime. This is hereinafter referred to as Assumption 3.

As previously discussed, DERs are presumed to be equipped with embeddedcontroller devices that drive the output powers to the commandedsetpoints; relevant dynamical models for the output powers of invertersoperating in a grid-connected mode can be found in datasheets ofcommercially available DERs and/or in the literature. If the timeconstant of the controllers is longer than h, Assumption 3 bounds thediscrepancy between the sampled output power and the commanded setpoint.For future developments, define the vector e^((k)):=[L+r_(p))e_(x)^((k)),1₂ ^(T)e_(ν) ^((k)),1₂ ^(T)e₀ ^((k)),e_(L) ^((k))]^(T), andnotice from Assumption 3 that ∥e^((k))∥₂≤e, e:=√{square root over((L+r_(p))²e_(x) ²+2e_(ν) ²+2e₀ ²+e_(L) ². )}

Let

$z^{(k)}:=\left\lbrack {\left( x^{(k)} \right)^{T},\left( {\overset{\_}{x}}^{(k)} \right)^{T},\left( d^{(k)} \right)^{T}} \right\rbrack^{T}$

collect the primal and dual variables produced by the real-timealgorithm at time t_(k). Based on Assumptions 1-3, the main convergenceresults are established next.

Consider the sequence {z^((k))} generated by the algorithm (18)-(29).The distance between z^((k)) and the primal-dual optimizer x^((k,*)) attime t_(k) can be bounded as:

∥z ^((k)) −z ^((k,*))∥₂ ≤c(α,r_(p) ,r _(d))^(k) ∥z ⁽⁰⁾ −z^((0,*))∥₂+Σ_(l=0) ^(k−1) c(α,r _(p) ,r _(d))^(l)(α∥e^((l))∥₂+σ^((l)))^(k−l−1)  (36)

where

c(α,r _(p) ,r _(d)):=[1−2α min{r _(p) ,r _(d)}+α²(L+r _(p)+5G)² +5α²(G+r _(d))²]^(½)  (37)

and σ^((k)) is defined in (30). This is hereinafter referred to asTheorem 5.

As a corollary, if c(α,r_(p),r_(d))<1, then the sequence {z^((k))}converges Q-linearly to {z^((k,*))} up to an asymptotic error boundgiven by:

$\begin{matrix}{{\limsup\limits_{k->\infty}{{z^{(k)} - z^{{({k,}}{*)}}}}_{2}} \leq \frac{\Delta}{1 - {c\left( {\alpha,r_{p},r_{d}} \right)}}} & (38)\end{matrix}$

where Δ:=αe+σ.

Notice first that the condition c(α,r_(p),r_(d))<1 is satisfied if

$\begin{matrix}{a < {\frac{\min \left\{ {r_{p},r_{d}} \right\}}{\left( {L + r_{p} + {5G}} \right)^{2} + {5\left( {G + r_{d}} \right)^{2}}}.}} & (39)\end{matrix}$

The bound (36) provides a characterization of the discrepancy betweenz^((k,*)) and z^((k)) at each time t_(k). On the other hand, theasymptotic bound (38) depends on the underlying dynamics of thedistribution system through σ and on the measurement errors through e.The result (38) can also be interpreted as input-to-state stability,where the optimal trajectory {z^((k,*))} of the time-varying problem(12) is taken as a reference. Interestingly, when e=0 and σ=0, thealgorithm converges to the solution of the static optimization problem(17).

As a conclusion, average tracking properties for the updates (25) and(29) should be established. To this end, some pertinent definitions andassumptions regarding DERs with discrete commands are introduced. TheVoronoi cell associated with a set χ ⊆

² and a point x ε χ is defined as ν_(χ)(x):={y ε

²:∥x−y∥≤∥x′−y∥, ∀x′ ε χ}. The following is then assumed.

Consider the collection of bounded Voronoi cells of χ_(j) ^((k)), k=1,2,. . . :

{ν_(χj) _((k)) (x):xεχ _(j) ^((k)),|ν_(χj) _((k)) (x)|<∞,k=1,2, . . . }

The sizes of these bounded Voronoi cells are uniformly bounded. This ishereinafter referred to as Assumption 4.

The collection {ch χ_(j) ^((k)), k=1,2, . . . } is a collection ofpolytopes such that:

-   -   i The sizes of the polytopes are uniformly bounded; and,    -   ii The set out outgoing normal to the faces of the polytopes is        finite.        This is hereinafter referred to as Assumption 5.

The following result establishes average tracking properties in terms ofimplementable setpoints for DERs with discrete control actions.

Under Assumptions 4 and 5, for each DER j with nonconvex operationalregion χ_(j) ^((k)) there exists a finite constant E_(j) such that∥e_(j) ^((k))∥₂≤E_(j) for all k. Consequently,

$\begin{matrix}{{{{\frac{1}{k}{\sum\limits_{ = 1}^{k}x_{j}^{()}}} - {\frac{1}{k}{\sum\limits_{ = 1}^{k}{\overset{\sim}{x}}_{j}^{()}}}}}_{2} \leq \frac{E_{j}}{k}} & (40)\end{matrix}$

and ∥x_(j) ^((k))−{tilde over (x)}_(j) ^((k))∥₂≤2E_(j) for all k. Thisis hereinafter referred to as Assumption 6.

FIG. 3 is a flow diagram illustrating example operations for performingreal time feedback-based optimization of distributed energy resources,in accordance with one or more aspects of the present disclosure. FIG. 3represents only one example process for performing real timefeedback-based optimization of distributed energy resources as describedherein, and various other or additional operations may be used in otherexamples. The example operations of FIG. 3 are described below withinthe context of FIG. 1.

In the example of FIG. 3, one or more nodes of a power system may beconfigured to measure, estimate, or otherwise determine node electricalquantities (100). For example, some of nodes 6 (e.g., voltage nodes) maybe configured to measure node voltages. Some of nodes 6 (e.g., currentnodes) may be configured to measure node currents. In other words, insome examples, each of nodes 6 may be (or include) voltage and/orcurrent measurement devices that are capable of measuring voltagesand/or currents at the respective node. Each of nodes 6 may determinerespective voltage or current measurements that correspond to thedifferent electrical phases at the node. In some examples, one or moreof control devices 10 and/or other components in the power system mayadditionally or alternatively be configured to measure voltages and/orcurrents at their respective locations.

The power system nodes, in the example of FIG. 3, may each output themeasured node electrical quantities (101). For instance, each of nodes 6may transmit its respective voltage or current measurements to a powermanagement unit via one or more wired or wireless communicationnetworks. In other examples, nodes 6 may additionally or alternativelytransmit the voltage and/or current measurements to one or more othercomponents within system 2, such as to one or more of control devices10, to one another, or to other components not shown in FIG. 2.

In the example of FIG. 3, a connection point of the power system (e.g.,the point at which the power system connects to a larger power system)may be configured to measure, estimate, or otherwise determineconnection point powers (103). For example, connection point 7 may be apower substation or other facility capable of determining the power flowat the connection point. Connection point 7 may determine powermeasurements that correspond to different electrical phases at theconnection point.

The connection point, in the example of FIG. 3, may output the measuredpowers (104). For instance, connection point 7 may transmit its powermeasurements to the power management unit via one or more wired orwireless communication networks. In some examples, connection point 7may additionally or alternatively transmit the power measurements to oneor more other components within system 2.

In the example of FIG. 3, a power management unit may receive the nodevoltages, node currents, and connection point powers (106). Powermanagement unit 4, for instance, may receive measurements 12 and powermeasurement 13.

For each voltage node, the power management unit may determine, based onthe plurality of node voltages, a respective value of a firstvoltage-constraint coefficient and a respective value of a secondvoltage-constraint coefficient (107). For instance, power managementunit 4 may determine, for each of nodes 6A and 6B (and possibly otherlocations), a respective value of the first voltage-constraintcoefficient based on a respective previous value of the firstvoltage-constraint coefficient, a minimum voltage value, and therespective voltage measurement for the voltage node. Power managementunit 4 may determine a respective value of the second voltage-constraintcoefficient based on a respective previous value for the secondvoltage-constraint coefficient, a maximum voltage value, and therespective voltage measurement. In some examples, power management unit4 may determine the respective values of the first and secondvoltage-constraint coefficients based additionally or alternatively onother criteria. In some examples, power management unit 4 may determinerespective values of the first and second voltage-constraintcoefficients for each phase at each voltage node. In other words, insome examples, power management unit 4 may determine values of the firstand second voltage-constraint coefficients for each voltage measurementreceived.

For each current node, the power management unit may determine, based onthe plurality of node currents, a respective value of acurrent-constraint coefficient (108). For instance, power managementunit 4 may determine, for node 6C (and possibly other locations), arespective value of the current-constraint coefficient based on arespective previous value of the current-constraint coefficient, amaximum current value, and the respective current measurement for thecurrent node. In some examples, power management unit 4 may determinethe respective values of the current-constraint coefficients basedadditionally or alternatively on other criteria. In some examples, powermanagement unit 4 may determine respective values of thecurrent-constraint coefficients for each phase at each current node. Inother words, in some examples, power management unit 4 may determinevalues of the current-constraint coefficient for each currentmeasurement received.

For the connection point, the power management unit may determine, basedon the power measurements, a value of a first power-constraintcoefficient and a respective value of a second power-constraintcoefficient (109). For instance, power management unit 4 may determine,for connection point 7 (and possibly other locations), a respectivevalue of the first power-constraint coefficient based on a respectiveprevious value of the first power-constraint coefficient, a powersetpoint value, and the respective power measurement. Power managementunit 4 may determine a respective value of the second power-constraintcoefficient based on a respective previous value for the secondpower-constraint coefficient, a power setpoint value, and the respectivepower measurement. In some examples, power management unit 4 maydetermine the respective values of the first and second power-constraintcoefficients based additionally or alternatively on other criteria. Insome examples, power management unit 4 may determine respective valuesof the first and second power-constraint coefficients for each phase atthe connection point. In other words, in some examples, power managementunit 4 may determine values of the first and second power-constraintcoefficients for each voltage measurement received.

In the example of FIG. 3, the power management unit may output therespective values of the first and second voltage-constraintcoefficients, the current-constraint coefficient, and the first andsecond power-constraint coefficients (110). For instance, powermanagement unit 4 may output coefficient values 14 to one or more othercomponents of network 2 via one or more wired or wireless communicationnetworks.

In the example of FIG. 3, one or more energy resource control devicesmay receive the coefficient values (111). Based on the value of thefirst and second voltage-constraint coefficients for each voltage node,the value of the current-constraint coefficients for each current node,and the value of the first and second power-constraint coefficients forthe connection point, the one or more energy resource control devicesmay determine a power setpoint value (112). For example, one or more ofcontrol devices 10 may receive coefficient values 14 and determine apower setpoint value for the corresponding energy resource or resources8 based on the coefficient values. In some examples, control devices 10may determine the power setpoint value based additionally oralternatively on other criteria, such as one or more performancemetrics. The performance metrics may be defined by a manager and/orowner of the corresponding energy resource(s) and/or by amanager/operator of the power system. Examples of performance metricsinclude a metric indicating cost for ancillary service provisioning, ametric indicating feed-in tariffs, and other metrics.

Based on the power setpoint value, the one or more energy resourcecontrol devices may, in the example of FIG. 3, modify an output power ofan associated energy resource (114). For example, control device 10A maymodify output powers of energy resource 8A. In various examples,modifying the output power may represent modifying a real output power,a reactive output power, or both.

The example operations of FIG. 3 may be performed in an iterativefashion. That is, while only a single flow is shown, each of operations100, 101, 103, 104, 106, 107, 108, 109, 110, 111, 112, and/or 114 may beperformed any number of times. In some examples, the operations may beperformed periodically. In some such examples, the frequency with whichthese operations are performed may be the same. In other such examples,one or more of the operations may be performed with higher or lowerfrequency than other operations.

Additionally, while shown in the example of FIG. 3 as being performed byspecific components, operations 100, 101, 103, 104, 106, 107, 108, 109,110, 111, 112, and/or 114 may, in other examples, be performed bycomponents other than those indicated. For instance, in some examplesoperations 106, 107, 108, 109, 110, 111, 112, and 114 may all beperformed by the power management unit. An additional variation of thetechniques described herein is described below with respect to FIG. 4.

FIG. 4 is a flow diagram illustrating example operations for regulatingvoltage through gather and broadcast techniques, in accordance with oneor more aspects of the present disclosure. FIG. 4 represents only oneexample process for performing real time feedback-based optimization ofdistributed energy resources as described herein, and various other oradditional operations may be used in other examples. The exampleoperations of FIG. 4 are described below within the context of FIG. 1.

In the example of FIG. 4, operations 200, 201, 203, 204, 206, 207, 208,209, and 210, are substantially the same as operations 100, 101, 103,104, 106, 107, 108, 109, and 110, respectively, as described withrespect to FIG. 3. In FIG. 4, however, power management unit 4 maytransmit coefficient values 14 to a control device that manages morethan one energy resource, such as control device 10B as shown in FIG. 1.

The aggregate energy resource manager may receive the coefficient values(211) and determine an aggregate power setpoint value based on thevalues of the first and second voltage-constraint coefficients, thevalues of the current-constraint coefficients, the values of the firstand second power-constraint coefficients, and a current aggregate outputpower (212). For instance, control device 10B may determine theaggregate power setpoint value based in part on the aggregate poweroutput of energy resources 8B, 8C, and 8D.

In the example of FIG. 3, the aggregate energy resource manager maydetermine individual power setpoint values based on the aggregate powersetpoint value (213). For example, control device 10B may determineindividual power setpoint values for each of energy resources 8B, 8C,and 8D. The aggregate energy resource manager may then modify the outputpower of each energy resource based on the respective individual powersetpoint value (214). For instance, control device 10B may individuallymodify the output power of energy resource 8B, energy resource 8C, andenergy resource 8D based on the individual power setpoint values.

As in FIG. 3, the example operations of FIG. 4 may be performed in aniterative fashion. Furthermore, while shown in the example of FIG. 4 asbeing performed by specific components, at least some of the operationsmay be performed by other components in system 2 or by other components.

FIG. 5 illustrates an additional example of a distribution systemconfigured to implement one or more techniques of the presentdisclosure. The techniques described herein may be used in the system ofFIG. 5 to address unique needs of utilities companies, aggregators, andDER-owners. The systems, devices, and methods of the present disclosuremay unify real-time voltage and frequency control at thehome/building/DER controllers' level with network-wide power managementat the utility/aggregator level. Additionally, these systems, devices,and methods may collectively enable feeders to emulate virtual powerplants providing services to the main grid at multiple temporal scales.The framework detailed herein offers responsiveness to rapidly changingconditions by incorporating intrinsic network physics into thecontrol-law formulation and processing real-time measurements.

Example implementation settings and devices are outlined below withrespect to the illustrative system in FIG. 5.

One example system includes a utility control platform. Such a systemmay include implementation of operations (a1), (c1), and (c2), as shownin FIG. 5. This system may offer an optimization and control platformfor utility companies/distribution system operators for real-timecontrol of utility-owned DERs deployed in feeders. The optimization andcontrol platform can be designed to pursue network-level objectives suchas voltage regulation, power loss minimization, and net-load smoothing,while respecting circuit laws, voltage limits, ramping constraints, andother limits. Responsiveness to high-level commands, such as thoseoriginating from an independent system operator, is also possible. Thecontrol platform can be utilized by utilitycompanies/distribution-system operators to also control customer-ownedDERs (e.g., provided that contractual agreements are in place). Thecontrol methods of such a system can be encoded into a software packageimplementable in various computing devices. Such a software packagecould afford a number of distinct installations, as exampled below.

In one example, existing Advanced Distribution Management Systems(ADMSs) may not support functionalities for real-time control of DERsand may typically send dispatch commands to the devices only at a slowtime scale (e.g., every 5 minutes, every 10 minutes, or at some otherfrequency). The systems, devices, and methods detailed herein may beembedded into such existing ADMSs to enable real-time control ofutility-controlled DERs (e.g., the DER power outputs can be adjusted onthe order of seconds, sub-seconds, etc.). The forecasting and stateestimation modules of existing ADMS may be interfaced with the controlsoftware to provide inputs to the controllers. The power commandsproduced by the control software can be sent to the DERs by leveragingexisting communication functionalities of the ADMS as well asstandardized communication protocols (e.g., DNP3 via TCP/IP, Modbus viaTCP/IP, and others). This framework can be extended to manageconventional devices, such as tap-changers and capacitor banks, inaddition to emerging DER types.

In another example, the control software for the system may be embeddedinto a standalone computational device (e.g., a control device) thatdirectly interacts with DERs. The control device may interact with theADMS (if deployed) to gather information regarding the distributionsystem state. Such communication with the ADMS may be standardized,e.g., via a new communication protocol. In some examples, emphasis maybe placed on compatibility of the control devices with existing ADMSstructures used by utilities.

Additionally or alternatively, the system's control platform can beenhanced to communicate directly with phasor measurement units to gathermeasurements, and to implement state estimation and forecastingalgorithms. Communication with DERs can be performed via standardizedprotocols (e.g., DNP3 via TCP/IP, Modbus via TCP/IP, etc.).

Another example system includes an aggregator control platform. Such asystem may also include implementation of operations (a1), (c1), and(c2) as shown in FIG. 5. This system may offer an optimization andcontrol platform for aggregators, to enable real-time control of DERsdeployed in a portion of the distribution system. Example portions ofthe distribution system include residential neighborhoods, communities,urban environments, and others. The aggregator control platform mayimplement the optimization and control methods described herein and mayinterface directly with the DERs via standardized communications. Insome examples, the control devices may be designed to pursue objectivessuch as voltage regulation, power loss minimization, and maximization ofrenewable utilization. Another example objective may be to respond tocommands sent from the utility company to regulate the power flows atthe aggregator-utility interface.

In such a system, control devices may interact with the ADMS (ifdeployed) to gather information regarding the distribution system state.In some examples, the communication protocol to communicate with theADMS may be standardized. In some examples, the control platform may beenhanced to implement state estimation and forecasting algorithms toimprove performance.

Another example system includes a distributed control platform. Such asystem may include implementation of operations (a1), (a2), (c1), (c2),and (c3) as shown in FIG. 5. In such a system, the examples above may becomplemented by a customer-level control device and a communication link(e.g., (c3)) between the customer-level control device and theutility/aggregator control device. The customer-level control device maymanage the behind-the-meter DERs based on, for example, 1) localobjective functions that are flexibly defined by end-users toaccommodate multiple interests and 2) global signals that arebroadcasted by the utility/aggregator control device (e.g., via (c3)) toincentivize DERs to adjust the power output to achieve global voltageregulation and ancillary service requests. The customer may be able toadjust and set a balance between these preferences via a user-interface.In some examples, in order to preserve privacy, customer's preferencesmay not be shared with the utility/aggregator.

The design of this system may better respect the computationalcapabilities of typical microcontrollers (e.g., as in micro-inverters)and may affords a low-cost hardware implementation. Various userpreferences may be specified within the scope of this disclosure, andimplementation of the techniques described herein may use existingmicrocontroller boards or custom-designed hardware.

The communication link (c3) may be designed based on standardizedprotocols (e.g., UDP) and may include cyber-protection applications toprevent cyberattacks. Depending on the goals of the specific system, thecustomer-level DER-control devices may be either commercially availabledevices (e.g., devices already used by current DER manufacturers thatare reconfigured), or, other computing devices, such as custom hardware.In the second case, the customer-level DER-control device maycommunicate with the DERs via standardized protocols (e.g., DNP3 viaTCP/IP, Modbus via TCP/IP, etc.).

In some instances, the three example systems described with respect toFIG. 5 may seamlessly coexist in the same distribution network. Thisdoes not preclude the existence of aggregators in circuits that deploy autility control platform and, vice versa, may allow utilities,aggregators, and end customers to pursue their own operationalobjectives, while naturally achieving global coordination to satisfyreliability constraints and enable feeders to provide services to themain grid.

The present disclosure allows for distribution networks featuringdistributed energy resources (DERs), and provides a system-theoreticcontrol framework that may allow distribution networks to emulatevirtual power plants effectively providing services at the interfacewith the transmission grid. The controller devices described herein mayadjust the output powers of individual DERs in response to automaticgeneration control and regulation services, while concurrentlyregulating voltages, currents, and powers within the feeder andmaximizing customers' and utility's performance objectives. The controlparadigm described herein may afford both a centralized and adistributed implementation, and their design is grounded on suitablelinearizations of the AC power-flow equations as well as onlineprimal-dual-type methods for time-varying optimization problems.Convergence and tracking capabilities of the provided controller deviceshave been analytically established under suitable modeling assumptions.Simulations have also been provided to validate the disclosed systems,devices, and methods.

At least some of the techniques of the present disclosure may beadditionally or alternatively described by one or more of the followingexamples.

Example 1

A device comprising: at least one processor configured to: receive aplurality of voltage values, wherein voltage values in the plurality ofvoltage values correspond to respective voltage nodes in a plurality ofvoltage nodes in a first portion of a power system; determine, for eachrespective voltage node: a respective value of a firstvoltage-constraint coefficient, based on a respective previous value ofthe first voltage-constraint coefficient, a minimum voltage value, and arespective voltage value in the plurality of voltage values thatcorresponds to the respective voltage node; and a respective value of asecond voltage-constraint coefficient based on a respective previousvalue of the second voltage-constraint coefficient, a maximum voltagevalue, and the respective voltage value; receive a power valuecorresponding to a connection point of the first portion of the powersystem with a second portion of the power system; determine for theconnection point: a value of a first power-constraint coefficient, basedon a previous value of the first power-constraint coefficient, a powersetpoint for the connection point, and the power value; and a value of asecond power-constraint coefficient based on a previous value of thesecond voltage-constraint coefficient, the power setpoint for theconnection point, and the power value; and cause at least one energyresource in a plurality of energy resources that are connected to thefirst portion of the power system to modify an output power of the atleast one energy resource based on the respective value of the firstvoltage-constraint coefficient for each respective voltage node, therespective value of the second voltage-constraint coefficient for eachrespective voltage node, the value of the first power-constraintcoefficient for the connection point, and the value of the secondpower-constraint coefficient for the connection point.

Example 2

The device of example 1, wherein: the at least one processor is furtherconfigured to: receive a plurality of current values, wherein currentvalues in the plurality of current values correspond to respectivecurrent nodes in a plurality of current nodes in the first portion ofthe power system; and determine, for each respective current node, arespective value of a current-constraint coefficient, based on arespective previous value of the current-constraint coefficient, arespective maximum current value for the respective current node, and arespective current value in the plurality of current values thatcorresponds to the respective current node, and causing the at least oneenergy resource to modify the output power of the at least one energyresource comprises causing the at least one energy resource to modifythe output power of the at least one energy resource based further onthe respective value of the current-constraint coefficient for eachrespective current node.

Example 3

The device of either of examples 1 or 2, wherein causing the at leastone energy resource to modify the output power comprises outputting, tothe at least one energy resource, the respective value of the firstvoltage-constraint coefficient for each respective voltage node, therespective value of the second voltage-constraint coefficient for eachrespective voltage node, the value of the first power-constraintcoefficient for the connection point, and the value of the secondpower-constraint coefficient for the connection point.

Example 4

The device of any of examples 1-3, wherein: each voltage value in theplurality of voltage values comprises a set of voltage values, eachcorresponding to a respective phase at the respective voltage node, andthe power value comprises a set of power values, each corresponding to arespective phase at the connection point.

Example 5

The device of example 4, wherein: determining the respective value ofthe first voltage-constraint coefficient comprises determining a set ofrespective values of the first voltage-constraint coefficient, eachrespective value in the set of respective values of the firstvoltage-constraint coefficient corresponding to the respective phase atthe respective voltage node; determining the respective value of thesecond voltage-constraint coefficient comprises determining a set ofrespective values of the second voltage-constraint coefficient, eachrespective value in the set of respective values of the secondvoltage-constraint coefficient corresponding to the respective phase atthe respective voltage node; determining the value of the firstpower-constraint coefficient comprises determining a set of respectivevalues of the first power-constraint coefficient, each respective valuein the set of respective values of the first power-constraintcoefficient corresponding to the respective phase at the connectionpoint; and determining the value of the second power-constraintcoefficient comprises determining a set of respective values of thesecond power-constraint coefficient, each respective value in the set ofrespective values of the second power-constraint coefficientcorresponding to the respective phase at the connection point.

Example 6

The device of any of examples 1-5, wherein: determining the respectivevalue of the first voltage-constraint coefficient comprises:determining, based on the respective previous value of the firstvoltage-constraint coefficient, the minimum voltage value, and therespective voltage value in the plurality of voltage values thatcorresponds to the respective voltage node, a respective first voltagecoefficient offset value; scaling the respective first voltagecoefficient offset value by a step size to determine a respective scaledfirst voltage coefficient offset value; responsive to determining that arespective first sum of the respective previous value of the firstvoltage-constraint coefficient and the respective scaled first voltagecoefficient offset value is greater than zero, setting the respectivevalue of the first voltage-constraint coefficient to be the respectivefirst sum; and responsive to determining that the respective first sumis less than or equal to zero, setting the respective value of the firstvoltage-constraint coefficient to be zero, determining the respectivevalue of the second voltage-constraint coefficient comprises:determining, based on the respective previous value of the secondvoltage-constraint coefficient, the maximum voltage value, and therespective voltage value in the plurality of voltage values thatcorresponds to the respective voltage node, a respective second voltagecoefficient offset value; scaling the respective second voltagecoefficient offset value by the step size to determine a respectivescaled second voltage coefficient offset value; responsive todetermining that a respective second sum of the respective previousvalue of the second voltage-constraint coefficient and the respectivescaled second voltage coefficient offset value is greater than zero,setting the respective value of the second voltage-constraintcoefficient to be the respective second sum; and responsive todetermining that the respective second sum is less than or equal tozero, setting the respective value of the second voltage-constraintcoefficient to be zero, determining the value of the firstpower-constraint coefficient comprises: determining, based on theprevious value of the first power-constraint coefficient, the powersetpoint for the connection point, an accuracy value, and the powervalue, a first power coefficient offset value; scaling the first powercoefficient offset value by the step size to determine a scaled firstpower coefficient offset value; and responsive to determining that athird sum of the previous value of the first power-constraintcoefficient and the scaled first power coefficient offset value isgreater than zero, setting the value of the first power-constraintcoefficient to be the third sum; and responsive to determining that thethird sum is less than or equal to zero, setting the value of the firstpower-constraint coefficient to be zero, and determining the value ofthe second power-constraint coefficient comprises: determining, based onthe previous value of the second power-constraint coefficient, the powersetpoint for the connection point, the accuracy value, and the powervalue, a second power coefficient offset value; scaling the second powercoefficient offset value by the step size to determine a scaled secondpower coefficient offset value; responsive to determining that a fourthsum of the previous value of the second power-constraint coefficient andthe scaled second power coefficient offset value is greater than zero,setting the value of the second power-constraint coefficient to be thefourth sum; and responsive to determining that the fourth sum is lessthan or equal to zero, setting the value of the second power-constraintcoefficient to be zero.

Example 7

The device of any of examples 1-6, wherein: determining the respectivevalue of the first voltage-constraint coefficient comprises calculating

{γ^((k))+α(ν^(min)−|{circumflex over (v)}^((tk))|−r_(d)γ^((k)))},wherein : γ^((k)) represents the respective previous value of the firstvoltage-constraint coefficient, ν^(min) represents the respectiveminimum voltage value for the respective voltage node, {circumflex over(v)}^((tk)) represents the respective voltage value that corresponds tothe respective voltage node, a represents a step size, and r_(d)represents a parameter indicating an importance of previous constraintcoefficient values, determining the respective value of the secondvoltage-constraint coefficient comprises calculating

proj_(ℝ₊){μ^((k)) + α(v̂^((tk)) − v^(ma x) − r_(d)μ^((k)))},

wherein: μ^((k)) represents the respective previous value of the secondvoltage-constraint coefficient, and ν^(max) represents the respectivemaximum voltage value for the respective voltage node, determining thevalue of the first power-constraint coefficient comprises calculating

proj_(ℝ₊){λ^((k)) + α(p̂₀^((k)) − p_(0, set)^((k)) − E^((k)) − r_(d)λ^((k)))},

wherein: λ^((k)) represents the previous value of the firstpower-constraint coefficient, p_(0,set) ^((k)) represents the powersetpoint for the connection point, {circumflex over (p)}₀ ^((k))represents the power value that corresponds to the connection point, andE^((k)) represents an accuracy value, and determining the value of thesecond power-constraint coefficient comprises calculating

proj_(ℝ₊³){v^((k)) + α(p_(0, set)^((k)) − p̂₀^((k)) − E^((k)) − r_(d)v^((k)))},

wherein: ν^((k)) represents the previous value of the secondpower-constraint coefficient.

Example 8

The device of any of examples 1-7, wherein causing the at least oneenergy resource to modify the output power comprises: determining, forthe at least one energy resource, a respective power setpoint value,based on the respective value of the first voltage-constraintcoefficient for each respective node, the respective value of the secondvoltage-constraint coefficient for each respective node, the value ofthe first power-constraint coefficient, the value of the secondpower-constraint coefficient, and a respective output power value of theat least one energy resource; and causing the at least one energyresource to modify the output power based on the respective powersetpoint.

Example 9

The device of any of examples 1-8, wherein: the at least one energyresource comprises an aggregation of energy resources, and causing theaggregation of energy resources to modify the output power comprises:determining, for the aggregation of energy resources, a respectiveaggregate power setpoint value, based on the respective value of thefirst voltage-constraint coefficient for each respective node, therespective value of the second voltage-constraint coefficient for eachrespective node, the value of the first power-constraint coefficient,the value of the second power-constraint coefficient, and a respectiveaggregate output power value of the aggregate energy resource;determining, based on the respective aggregate power setpoint value anda cost associated with each energy resource in the aggregation of energyresources, respective individual power setpoint values for each energyresource in the aggregation of energy resources; and causing each energyresource in the aggregation of energy resources to modify a respectiveoutput power based on the respective individual power setpoint values.

Example 10

A system comprising: a power management system configured to: receive,from each of a plurality of voltage measurement devices, a respectivevoltage value that corresponds to a respective voltage node in aplurality of voltage nodes in a first portion of a power system; receivea power value that corresponds to a connection point at which the firstportion of the power system connects to a second portion of the powersystem; determine, for each respective voltage node in the plurality ofvoltage nodes: a respective value of a first voltage-constraintcoefficient, based on a respective previous value of the firstvoltage-constraint coefficient, a minimum voltage value, and therespective voltage value; and a respective value of a secondvoltage-constraint coefficient based on a respective previous value ofthe second voltage-constraint coefficient, a maximum voltage value, andthe respective voltage value; determine, for the connection point: avalue of a first power-constraint coefficient, based on a previous valueof the first power-constraint coefficient, a power setpoint for theconnection point, and the power value; and a value of a secondpower-constraint coefficient, based on a previous value of the secondpower-constraint coefficient, the power setpoint for the connectionpoint, and the power value; and output the respective value of the firstvoltage-constraint coefficient for each respective voltage node, therespective value of the second voltage-constraint coefficient for eachrespective voltage node, the value of the first power-constraintcoefficient, and the value of the second power-constraint coefficient;and a plurality of energy resource management devices, eachcorresponding to a respective at least one energy resource connected tothe power system, each energy resource management device beingconfigured to: receive the respective value of the firstvoltage-constraint coefficient for each respective voltage node, therespective value of the second voltage-constraint coefficient for eachrespective voltage node, the value of the first power-constraintcoefficient, and the value of the second power-constraint coefficient;determine, based on the respective value of the first voltage-constraintcoefficient for each respective voltage node, the respective value ofthe second voltage-constraint coefficient for each respective voltagenode, the value of the first power-constraint coefficient, and the valueof the second power-constraint coefficient, a respective power setpointvalue; and modify a respective output power of the respective at leastone energy resource, based on the respective power setpoint value.

Example 11

The system of example 10, wherein the power management system is furtherconfigured to: receive, from each of a plurality of current measurementdevices, a respective current value that corresponds to a respectivecurrent node in a plurality of current nodes in the first portion of thepower system; determine, for each respective current node in theplurality of current nodes, a respective value of a current-constraintcoefficient, based on a respective previous value of thecurrent-constraint coefficient, a maximum current value, and therespective current value; and output the respective value of thecurrent-constraint coefficient for each respective current node, andeach energy resource management device is further configured to: receivethe respective value of the current-constraint coefficient for eachrespective current node; and determine the respective power setpointvalue based additionally on the respective value of thecurrent-constraint coefficient for each respective current node.

Example 12

The system of either of examples 10 or 11, wherein: the respectivevoltage value comprises a respective set of voltage values, eachcorresponding to a respective phase at the respective voltage node, andthe power value comprises a set of power values, each corresponding to arespective phase at the connection point.

Example 13

The system of example 12, wherein: determining the respective value ofthe first voltage-constraint coefficient comprises determining a set ofrespective values of the first voltage-constraint coefficient, eachrespective value in the set of respective values of the firstvoltage-constraint coefficient corresponding to the respective phase atthe respective voltage node; determining the respective value of thesecond voltage-constraint coefficient comprises determining a set ofrespective values of the second voltage-constraint coefficient, eachrespective value in the set of respective values of the secondvoltage-constraint coefficient corresponding to the respective phase atthe respective voltage node; determining the value of the firstpower-constraint coefficient comprises determining a set of respectivevalues of the first power-constraint coefficient, each respective valuein the set of respective values of the first power-constraintcoefficient corresponding to the respective phase at the connectionpoint; and determining the value of the second power-constraintcoefficient comprises determining a set of respective values of thesecond power-constraint coefficient, each respective value in the set ofrespective values of the second power-constraint coefficientcorresponding to the respective phase at the connection point.

Example 14

The system of any of examples 10-13, wherein a first energy resourcemanagement device in the plurality of energy resource management devicesmanages an aggregation of energy resources, the first energy resourcemanagement device being configured to: determine, based on therespective power setpoint value for the first energy resource managementdevice and a cost associated with each energy resource in theaggregation of energy resources, respective individual power setpointvalues for each energy resource in the aggregation of energy resources;and modify the respective output power of the aggregation of energyresources by modifying a respective individual output power of eachenergy resource in the aggregation of energy resources based on therespective individual power setpoint.

Example 15

The system of any of examples 10-14, wherein at least one of theplurality of energy resource management devices comprises a powerinverter that couples the respective at least one energy resource to thedistribution network.

Example 16

The system of any of examples 10-15, wherein: determining the respectivevalue of the first voltage-constraint coefficient comprises:determining, based on the respective previous value of the firstvoltage-constraint coefficient, the minimum voltage value, and therespective voltage value in the plurality of voltage values thatcorresponds to the respective voltage node, a respective first voltagecoefficient offset value; scaling the respective first voltagecoefficient offset value by a step size to determine a respective scaledfirst voltage coefficient offset value; responsive to determining that arespective first sum of the respective previous value of the firstvoltage-constraint coefficient and the respective scaled first voltagecoefficient offset value is greater than zero, setting the respectivevalue of the first voltage-constraint coefficient to be the respectivefirst sum; and responsive to determining that the respective first sumis less than or equal to zero, setting the respective value of the firstvoltage-constraint coefficient to be zero, determining the respectivevalue of the second voltage-constraint coefficient comprises:determining, based on the respective previous value of the secondvoltage-constraint coefficient, the maximum voltage value, and therespective voltage value in the plurality of voltage values thatcorresponds to the respective voltage node, a respective second voltagecoefficient offset value; scaling the respective second voltagecoefficient offset value by the step size to determine a respectivescaled second voltage coefficient offset value; responsive todetermining that a respective second sum of the respective previousvalue of the second voltage-constraint coefficient and the respectivescaled second voltage coefficient offset value is greater than zero,setting the respective value of the second voltage-constraintcoefficient to be the respective second sum; and responsive todetermining that the respective second sum is less than or equal tozero, setting the respective value of the second voltage-constraintcoefficient to be zero, determining the value of the firstpower-constraint coefficient comprises: determining, based on theprevious value of the first power-constraint coefficient, the powersetpoint for the connection point, an accuracy value, and the powervalue, a first power coefficient offset value; scaling the first powercoefficient offset value by the step size to determine a scaled firstpower coefficient offset value; and responsive to determining that athird sum of the previous value of the first power-constraintcoefficient and the scaled first power coefficient offset value isgreater than zero, setting the value of the first power-constraintcoefficient to be the third sum; and responsive to determining that thethird sum is less than or equal to zero, setting the value of the firstpower-constraint coefficient to be zero, and determining the value ofthe second power-constraint coefficient comprises: determining, based onthe previous value of the second power-constraint coefficient, the powersetpoint for the connection point, the accuracy value, and the powervalue, a second power coefficient offset value; scaling the second powercoefficient offset value by the step size to determine a scaled secondpower coefficient offset value; responsive to determining that a fourthsum of the previous value of the second power-constraint coefficient andthe scaled second power coefficient offset value is greater than zero,setting the value of the second power-constraint coefficient to be thefourth sum; and responsive to determining that the fourth sum is lessthan or equal to zero, setting the value of the second power-constraintcoefficient to be zero.

Example 17

The system of any of examples 10-16, further comprising the plurality ofvoltage measurement devices, each configured to: determine therespective voltage value; and output the respective voltage value.

Example 18

A method comprising: receiving, by a power management system comprisingat least one processor, a plurality of voltage values, wherein voltagevalues in the plurality of voltage values correspond to respectivevoltage nodes in a plurality of voltage nodes in a first portion of apower system; determining, by the power management system and for eachrespective voltage node: a respective value of a firstvoltage-constraint coefficient, based on a respective previous value ofthe first voltage-constraint coefficient, a minimum voltage value, and arespective voltage value in the plurality of voltage values thatcorresponds to the respective voltage node; and a respective value of asecond voltage-constraint coefficient based on a respective previousvalue of the second voltage-constraint coefficient, a maximum voltagevalue, and the respective voltage value; receiving, by the powermanagement system, a power value corresponding to a connection point ofthe first portion of the power system with a second portion of the powersystem; determining, by the power management system and for theconnection point: a value of a first power-constraint coefficient, basedon a previous value of the first power-constraint coefficient, a powersetpoint for the connection point, and the power value; and a value of asecond power-constraint coefficient based on a previous value of thesecond voltage-constraint coefficient, the power setpoint for theconnection point, and the power value; and causing, by the powermanagement system, at least one energy resource in a plurality of energyresources that are connected to the first portion of the power system tomodify an output power of the at least one energy resource based on therespective value of the first voltage-constraint coefficient for eachrespective voltage node, the respective value of the secondvoltage-constraint coefficient for each respective voltage node, thevalue of the first power-constraint coefficient for the connectionpoint, and the value of the second power-constraint coefficient for theconnection point.

Example 19

The method of example 18, wherein causing the at least one energyresource to modify the output power comprises outputting, to the atleast one energy resource, the respective value of the firstvoltage-constraint coefficient for each respective voltage node, therespective value of the second voltage-constraint coefficient for eachrespective voltage node, the value of the first power-constraintcoefficient for the connection point, and the value of the secondpower-constraint coefficient for the connection point.

Example 20

The method of either of examples 18 or 19, further comprising: receivinga plurality of current values, wherein current values in the pluralityof current values correspond to respective current nodes in a pluralityof current nodes in the first portion of the power system; determining,for each respective current node, a respective value of acurrent-constraint coefficient, based on a respective previous value ofthe current-constraint coefficient, a respective maximum current valuefor the respective current node, and a respective current value in theplurality of current values that corresponds to the respective currentnode, wherein causing the at least one energy resource to modify theoutput power of the at least one energy resource comprises causing theat least one energy resource to modify the output power of the at leastone energy resource based further on the respective value of thecurrent-constraint coefficient for each respective current node.

In one or more examples, the techniques described herein may beimplemented in hardware, software, firmware, or any combination thereofIf implemented in software, the functions may be stored on ortransmitted over, as one or more instructions or code, acomputer-readable medium and executed by a hardware-based processingunit. Computer-readable media may include computer-readable storagemedia, which corresponds to a tangible medium such as data storagemedia, or communication media, which includes any medium thatfacilitates transfer of a computer program from one place to another,e.g., according to a communication protocol. In this manner,computer-readable media generally may correspond to (1) tangiblecomputer-readable storage media, which is non-transitory or (2) acommunication medium such as a signal or carrier wave. Data storagemedia may be any available media that can be accessed by one or morecomputers or one or more processors to retrieve instructions, codeand/or data structures for implementation of the techniques described inthis disclosure. A computer program product may include acomputer-readable storage medium.

By way of example, and not limitation, such computer-readable storagemedia can comprise RAM, ROM, EEPROM, CD-ROM or other optical diskstorage, magnetic disk storage, or other magnetic storage devices, flashmemory, or any other medium that can be used to store desired programcode in the form of instructions or data structures and that can beaccessed by a computer. Also, any connection is properly termed acomputer-readable medium. For example, if instructions are transmittedfrom a website, server, or other remote source using a coaxial cable,fiber optic cable, twisted pair, digital subscriber line (DSL), orwireless technologies such as infrared, radio, and microwave, then thecoaxial cable, fiber optic cable, twisted pair, DSL, or wirelesstechnologies such as infrared, radio, and microwave are included in thedefinition of medium. It should be understood, however, thatcomputer-readable storage media and data storage media do not includeconnections, carrier waves, signals, or other transient media, but areinstead directed to non-transient, tangible storage media. Disk anddisc, as used herein, includes compact disc (CD), laser disc, opticaldisc, digital versatile disc (DVD), floppy disk and Blu-ray disc, wheredisks usually reproduce data magnetically, while discs reproduce dataoptically with lasers. Combinations of the above should also be includedwithin the scope of computer-readable media.

Instructions may be executed by one or more processors, such as one ormore digital signal processors (DSPs), general purpose microprocessors,application specific integrated circuits (ASICs), field programmablelogic arrays (FPGAs), or other equivalent integrated or discrete logiccircuitry. Accordingly, the term “processor,” as used herein may referto any of the foregoing structure or any other structure suitable forimplementation of the techniques described herein. In addition, in someaspects, the functionality described herein may be provided withindedicated hardware and/or software modules. Also, the techniques couldbe fully implemented in one or more circuits or logic elements.

The techniques of this disclosure may be implemented in a wide varietyof devices or apparatuses, including a wireless handset, an integratedcircuit (IC) or a set of ICs (e.g., a chip set). Various components,modules, or units are described in this disclosure to emphasizefunctional aspects of devices configured to perform the disclosedtechniques, but do not necessarily require realization by differenthardware units. Rather, as described above, various units may becombined in a hardware unit or provided by a collection ofinter-operative hardware units, including one or more processors asdescribed above, in conjunction with suitable software and/or firmware.

The foregoing disclosure includes various examples set forth merely asillustration. The disclosed examples are not intended to be limiting.Modifications incorporating the spirit and substance of the describedexamples may occur to persons skilled in the art. These and otherexamples are within the scope of this disclosure and the followingclaims.

What is claimed is:
 1. A device comprising: at least one processorconfigured to: receive a plurality of voltage values, wherein voltagevalues in the plurality of voltage values represent respective voltagemagnitudes at respective voltage nodes in a plurality of voltage nodesin a first portion of a power system; determine, for each respectivevoltage node: a respective value of a first voltage-constraintcoefficient, based on a respective previous value of the firstvoltage-constraint coefficient, a minimum voltage value, and arespective voltage value in the plurality of voltage values thatcorresponds to the respective voltage node; and a respective value of asecond voltage-constraint coefficient based on a respective previousvalue of the second voltage-constraint coefficient, a maximum voltagevalue, and the respective voltage value, wherein the minimum voltagevalue for the respective node and the maximum voltage value for therespective node represent a defined allowable voltage range at therespective node; receive a power value corresponding to a connectionpoint of the first portion of the power system with a second portion ofthe power system; determine for the connection point: a value of a firstpower-constraint coefficient, based on a previous value of the firstpower-constraint coefficient, a power setpoint for the connection point,and the power value; and a value of a second power-constraintcoefficient based on a previous value of the second voltage-constraintcoefficient, the power setpoint for the connection point, and the powervalue; and cause at least one energy resource in a plurality of energyresources that are connected to the first portion of the power system tomodify an output power of the at least one energy resource based on therespective value of the first voltage-constraint coefficient for eachrespective voltage node, the respective value of the secondvoltage-constraint coefficient for each respective voltage node, thevalue of the first power-constraint coefficient for the connectionpoint, and the value of the second power-constraint coefficient for theconnection point.
 2. The device of claim 1, wherein: the at least oneprocessor is further configured to: receive a plurality of currentvalues, wherein current values in the plurality of current valuescorrespond to respective current nodes in a plurality of current nodesin the first portion of the power system; and determine, for eachrespective current node, a respective value of a current-constraintcoefficient, based on a respective previous value of thecurrent-constraint coefficient, a respective maximum current value forthe respective current node, and a respective current value in theplurality of current values that corresponds to the respective currentnode, and causing the at least one energy resource to modify the outputpower of the at least one energy resource comprises causing the at leastone energy resource to modify the output power of the at least oneenergy resource based further on the respective value of thecurrent-constraint coefficient for each respective current node.
 3. Thedevice of claim 1, wherein causing the at least one energy resource tomodify the output power comprises outputting, to the at least one energyresource, the respective value of the first voltage-constraintcoefficient for each respective voltage node, the respective value ofthe second voltage-constraint coefficient for each respective voltagenode, the value of the first power-constraint coefficient for theconnection point, and the value of the second power-constraintcoefficient for the connection point.
 4. The device of claim 1, wherein:each voltage value in the plurality of voltage values comprises a set ofvoltage values, each corresponding to a respective phase at therespective voltage node, and the power value comprises a set of powervalues, each corresponding to a respective phase at the connectionpoint.
 5. The device of claim 4, wherein: determining the respectivevalue of the first voltage-constraint coefficient comprises determininga set of respective values of the first voltage-constraint coefficient,each respective value in the set of respective values of the firstvoltage-constraint coefficient corresponding to the respective phase atthe respective voltage node; determining the respective value of thesecond voltage-constraint coefficient comprises determining a set ofrespective values of the second voltage-constraint coefficient, eachrespective value in the set of respective values of the secondvoltage-constraint coefficient corresponding to the respective phase atthe respective voltage node; determining the value of the firstpower-constraint coefficient comprises determining a set of respectivevalues of the first power-constraint coefficient, each respective valuein the set of respective values of the first power-constraintcoefficient corresponding to the respective phase at the connectionpoint; and determining the value of the second power-constraintcoefficient comprises determining a set of respective values of thesecond power-constraint coefficient, each respective value in the set ofrespective values of the second power-constraint coefficientcorresponding to the respective phase at the connection point.
 6. Thedevice of claim 1, wherein: determining the respective value of thefirst voltage-constraint coefficient comprises: determining, based onthe respective previous value of the first voltage-constraintcoefficient, the minimum voltage value, and the respective voltage valuein the plurality of voltage values that corresponds to the respectivevoltage node, a respective first voltage coefficient offset value;scaling the respective first voltage coefficient offset value by a stepsize to determine a respective scaled first voltage coefficient offsetvalue; responsive to determining that a respective first sum of therespective previous value of the first voltage-constraint coefficientand the respective scaled first voltage coefficient offset value isgreater than zero, setting the respective value of the firstvoltage-constraint coefficient to be the respective first sum; andresponsive to determining that the respective first sum is less than orequal to zero, setting the respective value of the firstvoltage-constraint coefficient to be zero, determining the respectivevalue of the second voltage-constraint coefficient comprises:determining, based on the respective previous value of the secondvoltage-constraint coefficient, the maximum voltage value, and therespective voltage value in the plurality of voltage values thatcorresponds to the respective voltage node, a respective second voltagecoefficient offset value; scaling the respective second voltagecoefficient offset value by the step size to determine a respectivescaled second voltage coefficient offset value; responsive todetermining that a respective second sum of the respective previousvalue of the second voltage-constraint coefficient and the respectivescaled second voltage coefficient offset value is greater than zero,setting the respective value of the second voltage-constraintcoefficient to be the respective second sum; and responsive todetermining that the respective second sum is less than or equal tozero, setting the respective value of the second voltage-constraintcoefficient to be zero, determining the value of the firstpower-constraint coefficient comprises: determining, based on theprevious value of the first power-constraint coefficient, the powersetpoint for the connection point, an accuracy value, and the powervalue, a first power coefficient offset value; scaling the first powercoefficient offset value by the step size to determine a scaled firstpower coefficient offset value; and responsive to determining that athird sum of the previous value of the first power-constraintcoefficient and the scaled first power coefficient offset value isgreater than zero, setting the value of the first power-constraintcoefficient to be the third sum; and responsive to determining that thethird sum is less than or equal to zero, setting the value of the firstpower-constraint coefficient to be zero, and determining the value ofthe second power-constraint coefficient comprises: determining, based onthe previous value of the second power-constraint coefficient, the powersetpoint for the connection point, the accuracy value, and the powervalue, a second power coefficient offset value; scaling the second powercoefficient offset value by the step size to determine a scaled secondpower coefficient offset value; responsive to determining that a fourthsum of the previous value of the second power-constraint coefficient andthe scaled second power coefficient offset value is greater than zero,setting the value of the second power-constraint coefficient to be thefourth sum; and responsive to determining that the fourth sum is lessthan or equal to zero, setting the value of the second power-constraintcoefficient to be zero.
 7. The device of claim 1, wherein: determiningthe respective value of the first voltage-constraint coefficientcomprises calculating

{γ^((k))+α(ν^(min)−⊕{circumflex over (v)}^((tk))|−r_(d)γ^((k)))},wherein: γ^((k)) represents the respective previous value of the firstvoltage-constraint coefficient, ν^(min) represents the respectiveminimum voltage value for the respective voltage node, {circumflex over(v)}^((tk)) represents the respective voltage value that corresponds tothe respective voltage node, α represents a step size, and r_(d)represents a parameter indicating an importance of previous constraintcoefficient values, determining the respective value of the secondvoltage-constraint coefficient comprises calculating

{μ^((k))+α(|{circumflex over (v)}^((tk))|−ν^(max)−r_(d)μ^((k)))},wherein: μ^((k)) represents the respective previous value of the secondvoltage-constraint coefficient, and ν^(max) represents the respectivemaximum voltage value for the respective voltage node, determining thevalue of the first power-constraint coefficient comprises calculatingproj_(ℝ₊){λ^((k)) + α(p̂₀^((k)) − p_(0, set)^((k)) − E^((k)) − r_(d)λ^((k)))},wherein: λ^((k)) represents the previous value of the firstpower-constraint coefficient, p_(0,set) ^((k)) represents the powersetpoint for the connection point, {circumflex over (p)}₀ ^((k))represents the power value that corresponds to the connection point, andE^((k)) represents an accuracy value, and determining the value of thesecond power-constraint coefficient comprises calculatingproj_(ℝ₊³){v^((k)) + α(p_(0, set)^((k)) − p̂₀^((k)) − E^((k)) − r_(d)v^((k)))},wherein: ν^((k)) represents the previous value of the secondpower-constraint coefficient.
 8. The device of claim 1, wherein causingthe at least one energy resource to modify the output power comprises:determining, for the at least one energy resource, a respective powersetpoint value, based on the respective value of the firstvoltage-constraint coefficient for each respective node, the respectivevalue of the second voltage-constraint coefficient for each respectivenode, the value of the first power-constraint coefficient, the value ofthe second power-constraint coefficient, and a respective output powervalue of the at least one energy resource; and causing the at least oneenergy resource to modify the output power based on the respective powersetpoint.
 9. The device of claim 1, wherein: the at least one energyresource comprises an aggregation of energy resources, and causing theaggregation of energy resources to modify the output power comprises:determining, for the aggregation of energy resources, a respectiveaggregate power setpoint value, based on the respective value of thefirst voltage-constraint coefficient for each respective node, therespective value of the second voltage-constraint coefficient for eachrespective node, the value of the first power-constraint coefficient,the value of the second power-constraint coefficient, and a respectiveaggregate output power value of the aggregate energy resource;determining, based on the respective aggregate power setpoint value anda cost associated with each energy resource in the aggregation of energyresources, respective individual power setpoint values for each energyresource in the aggregation of energy resources; and causing each energyresource in the aggregation of energy resources to modify a respectiveoutput power based on the respective individual power setpoint values.10. A system comprising: a power management system configured to:receive, from each of a plurality of voltage measurement devices, arespective voltage value that represents a respective voltage magnitudeat a respective voltage node in a plurality of voltage nodes in a firstportion of a power system; receive a power value that corresponds to aconnection point at which the first portion of the power system connectsto a second portion of the power system; determine, for each respectivevoltage node in the plurality of voltage nodes: a respective value of afirst voltage-constraint coefficient, based on a respective previousvalue of the first voltage-constraint coefficient, a minimum voltagevalue, and the respective voltage value; and a respective value of asecond voltage-constraint coefficient based on a respective previousvalue of the second voltage-constraint coefficient, a maximum voltagevalue, and the respective voltage value, wherein the minimum voltagevalue for the respective node and the maximum voltage value for therespective node represent a defined allowable voltage range at therespective node; determine, for the connection point: a value of a firstpower-constraint coefficient, based on a previous value of the firstpower-constraint coefficient, a power setpoint for the connection point,and the power value; and a value of a second power-constraintcoefficient, based on a previous value of the second power-constraintcoefficient, the power setpoint for the connection point, and the powervalue; and output the respective value of the first voltage-constraintcoefficient for each respective voltage node, the respective value ofthe second voltage-constraint coefficient for each respective voltagenode, the value of the first power-constraint coefficient, and the valueof the second power-constraint coefficient; and a plurality of energyresource management devices, each corresponding to a respective at leastone energy resource connected to the power system, each energy resourcemanagement device being configured to: receive the respective value ofthe first voltage-constraint coefficient for each respective voltagenode, the respective value of the second voltage-constraint coefficientfor each respective voltage node, the value of the firstpower-constraint coefficient, and the value of the secondpower-constraint coefficient; determine, based on the respective valueof the first voltage-constraint coefficient for each respective voltagenode, the respective value of the second voltage-constraint coefficientfor each respective voltage node, the value of the firstpower-constraint coefficient, and the value of the secondpower-constraint coefficient, a respective power setpoint value; andmodify a respective output power of the respective at least one energyresource, based on the respective power setpoint value.
 11. The systemof claim 10, wherein the power management system is further configuredto: receive, from each of a plurality of current measurement devices, arespective current value that corresponds to a respective current nodein a plurality of current nodes in the first portion of the powersystem; determine, for each respective current node in the plurality ofcurrent nodes, a respective value of a current-constraint coefficient,based on a respective previous value of the current-constraintcoefficient, a maximum current value, and the respective current value;and output the respective value of the current-constraint coefficientfor each respective current node, and each energy resource managementdevice is further configured to: receive the respective value of thecurrent-constraint coefficient for each respective current node; anddetermine the respective power setpoint value based additionally on therespective value of the current-constraint coefficient for eachrespective current node.
 12. The system of claim 10, wherein: therespective voltage value comprises a respective set of voltage values,each corresponding to a respective phase at the respective voltage node,and the power value comprises a set of power values, each correspondingto a respective phase at the connection point.
 13. The system of claim12, wherein: determining the respective value of the firstvoltage-constraint coefficient comprises determining a set of respectivevalues of the first voltage-constraint coefficient, each respectivevalue in the set of respective values of the first voltage-constraintcoefficient corresponding to the respective phase at the respectivevoltage node; determining the respective value of the secondvoltage-constraint coefficient comprises determining a set of respectivevalues of the second voltage-constraint coefficient, each respectivevalue in the set of respective values of the second voltage-constraintcoefficient corresponding to the respective phase at the respectivevoltage node; determining the value of the first power-constraintcoefficient comprises determining a set of respective values of thefirst power-constraint coefficient, each respective value in the set ofrespective values of the first power-constraint coefficientcorresponding to the respective phase at the connection point; anddetermining the value of the second power-constraint coefficientcomprises determining a set of respective values of the secondpower-constraint coefficient, each respective value in the set ofrespective values of the second power-constraint coefficientcorresponding to the respective phase at the connection point.
 14. Thesystem of claim 10, wherein a first energy resource management device inthe plurality of energy resource management devices manages anaggregation of energy resources, the first energy resource managementdevice being configured to: determine, based on the respective powersetpoint value for the first energy resource management device and acost associated with each energy resource in the aggregation of energyresources, respective individual power setpoint values for each energyresource in the aggregation of energy resources; and modify therespective output power of the aggregation of energy resources bymodifying a respective individual output power of each energy resourcein the aggregation of energy resources based on the respectiveindividual power setpoint.
 15. The system of claim 10, wherein at leastone of the plurality of energy resource management devices comprises apower inverter that couples the respective at least one energy resourceto the distribution network.
 16. The system of claim 10, wherein:determining the respective value of the first voltage-constraintcoefficient comprises: determining, based on the respective previousvalue of the first voltage-constraint coefficient, the minimum voltagevalue, and the respective voltage value in the plurality of voltagevalues that corresponds to the respective voltage node, a respectivefirst voltage coefficient offset value; scaling the respective firstvoltage coefficient offset value by a step size to determine arespective scaled first voltage coefficient offset value; responsive todetermining that a respective first sum of the respective previous valueof the first voltage-constraint coefficient and the respective scaledfirst voltage coefficient offset value is greater than zero, setting therespective value of the first voltage-constraint coefficient to be therespective first sum; and responsive to determining that the respectivefirst sum is less than or equal to zero, setting the respective value ofthe first voltage-constraint coefficient to be zero, determining therespective value of the second voltage-constraint coefficient comprises:determining, based on the respective previous value of the secondvoltage-constraint coefficient, the maximum voltage value, and therespective voltage value in the plurality of voltage values thatcorresponds to the respective voltage node, a respective second voltagecoefficient offset value; scaling the respective second voltagecoefficient offset value by the step size to determine a respectivescaled second voltage coefficient offset value; responsive todetermining that a respective second sum of the respective previousvalue of the second voltage-constraint coefficient and the respectivescaled second voltage coefficient offset value is greater than zero,setting the respective value of the second voltage-constraintcoefficient to be the respective second sum; and responsive todetermining that the respective second sum is less than or equal tozero, setting the respective value of the second voltage-constraintcoefficient to be zero, determining the value of the firstpower-constraint coefficient comprises: determining, based on theprevious value of the first power-constraint coefficient, the powersetpoint for the connection point, an accuracy value, and the powervalue, a first power coefficient offset value; scaling the first powercoefficient offset value by the step size to determine a scaled firstpower coefficient offset value; and responsive to determining that athird sum of the previous value of the first power-constraintcoefficient and the scaled first power coefficient offset value isgreater than zero, setting the value of the first power-constraintcoefficient to be the third sum; and responsive to determining that thethird sum is less than or equal to zero, setting the value of the firstpower-constraint coefficient to be zero, and determining the value ofthe second power-constraint coefficient comprises: determining, based onthe previous value of the second power-constraint coefficient, the powersetpoint for the connection point, the accuracy value, and the powervalue, a second power coefficient offset value; scaling the second powercoefficient offset value by the step size to determine a scaled secondpower coefficient offset value; responsive to determining that a fourthsum of the previous value of the second power-constraint coefficient andthe scaled second power coefficient offset value is greater than zero,setting the value of the second power-constraint coefficient to be thefourth sum; and responsive to determining that the fourth sum is lessthan or equal to zero, setting the value of the second power-constraintcoefficient to be zero.
 17. The system of claim 10, further comprisingthe plurality of voltage measurement devices, each configured to:determine the respective voltage value; and output the respectivevoltage value.
 18. A method comprising: receiving, by a power managementsystem comprising at least one processor, a plurality of voltage values,wherein voltage values in the plurality of voltage values representrespective voltage magnitudes at respective voltage nodes in a pluralityof voltage nodes in a first portion of a power system; determining, bythe power management system and for each respective voltage node: arespective value of a first voltage-constraint coefficient, based on arespective previous value of the first voltage-constraint coefficient, aminimum voltage value, and a respective voltage value in the pluralityof voltage values that corresponds to the respective voltage node; and arespective value of a second voltage-constraint coefficient based on arespective previous value of the second voltage-constraint coefficient,a maximum voltage value, and the respective voltage value, wherein theminimum voltage value for the respective node and the maximum voltagevalue for the respective node represent a defined allowable voltagerange at the respective node; receiving, by the power management system,a power value corresponding to a connection point of the first portionof the power system with a second portion of the power system;determining, by the power management system and for the connectionpoint: a value of a first power-constraint coefficient, based on aprevious value of the first power-constraint coefficient, a powersetpoint for the connection point, and the power value; and a value of asecond power-constraint coefficient based on a previous value of thesecond voltage-constraint coefficient, the power setpoint for theconnection point, and the power value; and causing, by the powermanagement system, at least one energy resource in a plurality of energyresources that are connected to the first portion of the power system tomodify an output power of the at least one energy resource based on therespective value of the first voltage-constraint coefficient for eachrespective voltage node, the respective value of the secondvoltage-constraint coefficient for each respective voltage node, thevalue of the first power-constraint coefficient for the connectionpoint, and the value of the second power-constraint coefficient for theconnection point.
 19. The method of claim 18, wherein causing the atleast one energy resource to modify the output power comprisesoutputting, to the at least one energy resource, the respective value ofthe first voltage-constraint coefficient for each respective voltagenode, the respective value of the second voltage-constraint coefficientfor each respective voltage node, the value of the firstpower-constraint coefficient for the connection point, and the value ofthe second power-constraint coefficient for the connection point. 20.The method of claim 18, further comprising: receiving a plurality ofcurrent values, wherein current values in the plurality of currentvalues correspond to respective current nodes in a plurality of currentnodes in the first portion of the power system; determining, for eachrespective current node, a respective value of a current-constraintcoefficient, based on a respective previous value of thecurrent-constraint coefficient, a respective maximum current value forthe respective current node, and a respective current value in theplurality of current values that corresponds to the respective currentnode, wherein causing the at least one energy resource to modify theoutput power of the at least one energy resource comprises causing theat least one energy resource to modify the output power of the at leastone energy resource based further on the respective value of thecurrent-constraint coefficient for each respective current node.